COIN-OR IPOPT (Interior Point Optimizer) is an open-source solver for large-scale nonlinear programming (NLP). The code has been written primarily by Andreas Wächter.

IPOPT implements an interior point line search filter method for nonlinear programming models which functions can be nonconvex, but should be twice continuously differentiable. For more information on the algorithm we refer to [143, 192, 193, 194, 195] and the IPOPT documentation. Most of the IPOPT documentation in the section was taken from the IPOPT manual [103] .

Available linear solvers

The performance and robustness of IPOPT on larger models heavily relies on the used solver for sparse symmetric indefinite linear systems.

GAMS/IPOPT includes the sparse solver MUMPS [7, 8] (currently the default), and MKL PARDISO [165, 166]. The latter is not available for systems on ARM64 CPUs. In the commerically licensed GAMS/IPOPTH version, also the Harwell Subroutine Library (HSL) solvers MA27, MA57, HSL_MA86, and HSL_MA97 are available and MA27 is used by default.

MUMPS, MA57, HSL_MA86, and HSL_MA97 use METIS for matrix ordering [102], see also the METIS manual . METIS is copyrighted by the regents of the University of Minnesota.

IPOPT and IPOPTH can exploit parallelization of the linear solvers MKL Pardiso, HSL MA86, and HSL MA97 and the linear algebra routines (see next section).

The linear solver is chosen by the linear_solver option. Benchmarks have shown that MA57 and HSL_MA97 are often able to outperform MA27 on larger instances. Further, PARDISO often allows for performance that is better than MUMPS and similar to the HSL solvers. If IPOPT fails to solve an instance with PARDISO, it's worth to try changing the options pardisomkl_order and pardisomkl_max_iterative_refinement_steps.

It is also possible to use the linear solver PARDISO from the PARDISO Solver Project or the HSL routines with GAMS/IPOPT if a user provides libraries that can be loaded at runtime. PARDISO from the PARDISO Solver Project can provide performance that exceeds the one of PARDISO from Intel MKL. To build the HSL routines, COIN-OR project ThirdParty-HSL may be useful. See also options linear_solver, linear_system_scaling, nlp_scaling_method, pardisolib, and hsllib. Note that it is your responsibility to ensure that you are entitled to download and use these routines!

The linear algebra library

On systems for AMD and Intel CPUs, the IPOPT library distributed by GAMS and most of the linear solvers used by IPOPT use the Intel oneAPI Math Kernel Library (MKL), which provides a fast and parallel implementation of linear algebra routines (BLAS/LAPACK) and the linear solver PARDISO. MKL chooses an internal code path that provides the best possible performance for the used CPU type. As a consequence, results can be different when changing from one CPU to another. By setting an environment variable, the code path to use can be set by the user. See the Intel MKL documentation regarding Conditional Numerical Reproducibility for more details.

Intel MKL has been optimized for use with Intel CPUs. On CPUs from other vendors, MKL may not use an internal code path that could provide a better performance on that CPU. For example, it may not use AVX2 instructions on an AMD CPU that provides AVX2. However, Intel recently started to add optimized code for AMD's Zen CPUs.

To gain more insight into the use of MKL in GAMS/IPOPT, one may set the environment variable MKL_VERBOSE to 1. This will print out information about the MKL library used, functions being called, time spend there, etc.

On the GAMS system for macOS on ARM64 CPUs, the Apple Accelerate framework is used as linear algebra library.

Usage

The following statement can be used inside your GAMS program to specify using IPOPT

Option NLP = IPOPT;     { or LP, RMIP, DNLP, RMINLP, QCP, RMIQCP, CNS }

The above statement should appear before the Solve statement. If IPOPT was specified as the default solver during GAMS installation, the above statement is not necessary.

To use IPOPTH, the statement should be

Option NLP = IPOPTH;    { or LP, RMIP, DNLP, RMINLP, QCP, RMIQCP, CNS }

Specification of Options

IPOPT has many options that can be adjusted for the algorithm (see Section List of IPOPT Options). Options are all identified by a string name, and their values can be of one of three types: Number (real), Integer, or String. Number options are used for things like tolerances, integer options are used for things like maximum number of iterations, and string options are used for setting algorithm details, like the NLP scaling method. Options can be set by creating a ipopt.opt file in the directory you are executing IPOPT.

The ipopt.opt file is read line by line and each line should contain the option name, followed by whitespace, and then the value. Comments can be included with the # symbol. For example, the following is a valid ipopt.opt file:

# This is a comment

# Turn off the NLP scaling
nlp_scaling_method none

# Change the initial barrier parameter
mu_init 1e-2

# Set the max number of iterations
max_iter 500

GAMS/IPOPT understands currently the following GAMS parameters: reslim (time limit), iterlim (iteration limit), domlim (domain violation limit). Further the option threads can be used to control the number of threads used in the linear algebra routines and the linear solver. Setting threads=0 currently does not enable multithreaded linear algebra.

Warmstarting IPOPT

As an interior point solver, it is difficult to warm start IPOPT. By default, only the level values of the variables are passed as starting point to IPOPT. Setting the IPOPT option warm_start_init_point to yes enables that also dual values for variables and constraints are passed to IPOPT.

However, the expected behavior that IPOPT finishes within one iteration if optimal primal and dual values are passed is not reached this way, yet. This is, because IPOPT by default moves any initial value that is close to a bound into the interior. The amount on how much the initial point is moved can be controlled by various bound_push and bound_frac options. To make IPOPT accept an optimal primal/dual solution within one iteration, it should be sufficient to set the following options:

warm_start_init_point       yes
warm_start_bound_push       1e-9
warm_start_bound_frac       1e-9
warm_start_slack_bound_frac 1e-9
warm_start_slack_bound_push 1e-9
warm_start_mult_bound_push  1e-9

Further, it can be useful to specify that IPOPT can use a less central path in its first iterations by reducing the value of option mu_init. This option is only used if option mu_strategy is set to "monotone", so the option file entries would be

mu_strategy monotone
mu_init     0.0001

Finally, IPOPT by default checks whether it should scale the problem. The computed scaling depends on the starting point, which can be undesired when warmstarting. Thus, it may be useful to turn off scaling via option nlp_scaling_method:

nlp_scaling_method none

If a modified but structually equivalent problem instance is solved, e.g., via GUSS, option warm_start_init_point is automatically set to "yes" for every solve following the first one. If this is not desired, warm_start_init_point should explicitly be set to "no" in an IPOPT options file.

Output

This section describes the standard IPOPT console output. The output is designed to provide a quick summary of each iteration as IPOPT solves the problem.

Before IPOPT starts to solve the problem, it displays the problem statistics (number of nonzero-elements in the matrices, number of variables, etc.). Note that if you have fixed variables (both upper and lower bounds are equal), IPOPT may remove these variables from the problem internally and not include them in the problem statistics.

Following the problem statistics, IPOPT will begin to solve the problem and you will see output resembling the following,

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.6109693e+01 1.12e+01 5.28e-01   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.8029749e+01 9.90e-01 6.62e+01   0.1 2.05e+00    -  2.14e-01 1.00e+00f  1
   2  1.8719906e+01 1.25e-02 9.04e+00  -2.2 5.94e-02   2.0 8.04e-01 1.00e+00h  1

and the columns of output are defined as

item

The current iteration count. This includes regular iterations and iterations while in restoration phase. If the algorithm is in the restoration phase, the letter r will be appended to the iteration number.

objective

The unscaled objective value at the current point. During the restoration phase, this value remains the unscaled objective value for the original problem.

inf_pr

The unscaled constraint violation at the current point. This quantity is the infinity-norm (max) of the (unscaled) constraint violation. During the restoration phase, this value remains the constraint violation of the original problem at the current point. The option inf_pr_output can be used to switch to the printing of a different quantity. During the restoration phase, this value is the primal infeasibility of the original problem at the current point.

inf_du

The scaled dual infeasibility at the current point. This quantity measure the infinity-norm (max) of the internal dual infeasibility (Eq. (4a) in [194]), including inequality constraints reformulated using slack variables and problem scaling. During the restoration phase, this is the value of the dual infeasibility for the restoration phase problem.

lg(mu)

log₁₀ of the value of the barrier parameter μ.

||d||

The infinity norm (max) of the primal step (for the original variables x and the internal slack variables s). During the restoration phase, this value includes the values of additional variables, p and n in Eq. (10) of [194] .

lg(rg)

log₁₀ of the value of the regularization term for the Hessian of the Lagrangian in the augmented system ( \(\delta_w\) in Eq. (26) of [194]). A dash (-) indicates that no regularization was done.

alpha_du

The stepsize for the dual variables ( \(\alpha^z_k\) in Eq. (14c) of [194]).

alpha_pr

The stepsize for the primal variables ( \(\alpha_k\) in Eq. (14a) of [194]). The number is usually followed by a character for additional diagnostic information regarding the step acceptance criterion:

• f: f-type iteration in the filter method w/o second order correction
• F: f-type iteration in the filter method w/ second order correction
• h: h-type iteration in the filter method w/o second order correction
• H: h-type iteration in the filter method w/ second order correction
• k: penalty value unchanged in merit function method w/o second order correction
• K: penalty value unchanged in merit function method w/ second order correction
• n: penalty value updated in merit function method w/o second order correction
• N: penalty value updated in merit function method w/ second order correction
• R: Restoration phase just started
• w: in watchdog procedure
• s: step accepted in soft restoration phase
• t/T: tiny step accepted without line search
• r: some previous iterate restored

ls

The number of backtracking line search steps (does not include second-order correction steps).

Note that the step acceptance mechanisms in IPOPT consider the barrier objective function (Eq. (3a) in [194]) which is usually different from the value reported in the objective column. Similarly, for the purposes of the step acceptance, the constraint violation is measured for the internal problem formulation, which includes slack variables for inequality constraints and potentially scaling of the constraint functions. This value, too, is usually different from the value reported in inf_pr. As a consequence, a new iterate might have worse values both for the objective function and the constraint violation as reported in the iteration output, seemingly contradicting globalization procedure.

When the algorithm terminates, IPOPT will output a message to the screen. The following is a list of the possible output messages and a brief description.

Optimal Solution Found.

This message indicates that IPOPT found a (locally) optimal point within the desired tolerances.

Solved To Acceptable Level.

This indicates that the algorithm did not converge to the ''desired'' tolerances, but that it was able to obtain a point satisfying the ''acceptable'' tolerance level as specified by acceptable-* options. This may happen if the desired tolerances are too small for the current problem.

Feasible point for square problem found.

This message is printed if the problem is "square" (i.e., it has as many equality constraints as free variables) and IPOPT found a point that is feasible w.r.t. constr_viol_tol. It may, however, not be feasible w.r.t. tol.

Converged to a point of local infeasibility. Problem may be infeasible.

The restoration phase converged to a point that is a minimizer for the constraint violation (in the \(\ell_1\)-norm), but is not feasible for the original problem. This indicates that the problem may be infeasible (or at least that the algorithm is stuck at a locally infeasible point). The returned point (the minimizer of the constraint violation) might help you to find which constraint is causing the problem. If you believe that the NLP is feasible, it might help to start the optimization from a different point.

Search Direction is becoming Too Small.

This indicates that IPOPT is calculating very small step sizes and making very little progress. This could happen if the problem has been solved to the best numerical accuracy possible given the current scaling.

Iterates divering; problem might be unbounded.

This message is printed if the max-norm of the iterates becomes larger than the value of the option diverging_iterates_tol. This can happen if the problem is unbounded below and the iterates are diverging.

Stopping optimization at current point as requested by user.

This message is printed if either an interrupt signal was received (e.g., Ctrl+C was pressed) or the domain violation limit is reached.

Maximum Number of Iterations Exceeded.

This indicates that IPOPT has exceeded the maximum number of iterations as specified by the IPOPT option max_iter or the GAMS option iterlim.

Maximum wallclock time exceeded.

This indicates that IPOPT has exceeded the maximum number of wallclock seconds as specified by the IPOPT option max_wall_time or the GAMS option reslim.

Maximum CPU time exceeded.

This indicates that IPOPT has exceeded the maximum number of CPU seconds as specified by the IPOPT option max_cpu_time.

Restoration Failed!

This indicates that the restoration phase failed to find a feasible point that was acceptable to the filter line search for the original problem. This could happen if the problem is highly degenerate, does not satisfy the constraint qualification, or if an external or extrinsic function in GAMS provides incorrect derivative information.

Error in step computation!

This messages is printed if IPOPT is unable to compute a step towards a new iterate and the current iterate is not acceptable for the specified tolerances.

A possible reason is that a search direction could not be computed despite several attempts to modify the iteration matrix. Usually, the value of the regularization parameter then becomes too large.

Another reason is that the feasibility restoration phase could not be activated because the current iterate is not infeasible. Reasons for this again include that the problem is highly degenerate, badly scaled, or does not satisfy the constraint qualification. Before IPOPT 3.14, this resulted in a Restoration_Failed status code with message "Restoration phase is called at almost feasible point...".

Problem has too few degrees of freedom.

This indicates that your problem, as specified, has too few degrees of freedom. This can happen if you have too many equality constraints, or if you fix too many variables (IPOPT removes fixed variables by default, see also the option fixed_variable_treatment).

Not enough memory.

An error occurred while trying to allocate memory. The problem may be too large for your current memory and swap configuration. Sometimes it can help to choose a different linear solver.

INTERNAL ERROR: Unknown SolverReturn value - Notify IPOPT Authors.

An unknown internal error has occurred. Please notify the authors of the GAMS/IPOPT link or IPOPT (refer to support@gams.com).

Diagnostic Tags for IPOPT

To print additional diagnostic tags for each iteration of IPOPT, set the options print_info_string to yes. With this, a tag will appear at the end of an iteration line with the following diagnostic meaning that are useful to flag difficulties for a particular IPOPT run. The following is a list of possible strings:

• !: Tighten resto tolerance if only slightly infeasible, see Sec. 3.3 in [194]
• A: Current iteration is acceptable (alternate termination)
• a: Perturbation for PD Singularity can't be done, assume singular, see Sec. 3.1 in [194]
• C: Second Order Correction taken, see Sec. 2.4 in [194]
• Dh: Hessian degenerate based on multiple iterations, see Sec. 3.1 in [194]
• Dhj: Hessian/Jacobian degenerate based on multiple iterations, see Sec. 3.1 in [194]
• Dj: Jacobian degenerate based on multiple iterations, see Sec. 3.1 in [194]
• dx: \(\delta_x\) perturbation too large, see Sec. 3.1 in [194]
• e: Cutting back α due to evaluation error (in backtracking line search)
• F-: Filter should be reset, but maximal resets exceeded, see Sec. 2.3 in [194]
• F+: Resetting filter due to last few rejections of filter, see Sec. 2.3 in [194]
• L: Degenerate Jacobian, \(\delta_c\) already perturbed, see Sec. 3.1 in [194]
• l: Degenerate Jacobian, \(\delta_c\) perturbed, see Sec. 3.1 in [194]
• M: Magic step taken for slack variables (in backtracking line search)
• Nh: Hessian not yet degenerate, see Sec. 3.1 in [194]
• Nhj: Hessian/Jacobian not yet degenerate, see Sec. 3.1 in [194]
• Nj: Jacobian not yet degenerate, see Sec. 3.1 in [194]
• NW: Warm start initialization failed (in Warm Start Initialization)
• q: PD system possibly singular, attempt to improve solution quality, see Sec. 3.1 in [194]
• R: Solution of restoration phase, see Sec. 3.3 in [194]
• S: PD system possibly singular, accept current solution, see Sec. 3.1 in [194]
• s: PD system singular, see Sec. 3.1 in [194]
• s: Square Problem. Set multipliers to zero (default initialization routine)
• Tmax: Trial θ is larger than θ_max (filter parameter, Eq. (21) in [194])
• W: Watchdog line search procedure successful, see Sec. 3.2 in [194]
• w: Watchdog line search procedure unsuccessful, stopped, see Sec. 3.2 in [194]
• Wb: Undoing most recent SR1 update, see Sec. 5.4.1 in [21]
• We: Skip Limited-Memory Update in restoration phase, see Sec. 5.4.1 in [21]
• Wp: Safeguard \(B^0 = \sigma I\) for Limited-Memory Update, see Sec. 5.4.1 in [21]
• Wr: Resetting Limited-Memory Update, see Sec. 5.4.1 in [21]
• Ws: Skip Limited-Memory Update since \(s^Ty\) is not positive, see Sec. 5.4.1 in [21]
• WS: Skip Limited-Memory Update since \(\Delta x\) is too small, see Sec. 5.4.1 in [21]
• y: Dual infeasibility, use least square multiplier update (during IPOPT algorithm)
• z: Apply correction to bound multiplier if too large (during IPOPT algorithm)

List of IPOPT Options

Termination

Option	Description	Default
acceptable_compl_inf_tol	"Acceptance" threshold for the complementarity conditions.	0.01
acceptable_constr_viol_tol	"Acceptance" threshold for the constraint violation.	0.01
acceptable_dual_inf_tol	"Acceptance" threshold for the dual infeasibility.	1e+10
acceptable_iter	Number of "acceptable" iterates before triggering termination.	0
acceptable_obj_change_tol	"Acceptance" stopping criterion based on objective function change.	1e+20
acceptable_tol	"Acceptable" convergence tolerance (relative).	1e-06
compl_inf_tol	Desired threshold for the complementarity conditions.	0.0001
constr_viol_tol	Desired threshold for the constraint and variable bound violation.	1e-06
diverging_iterates_tol	Threshold for maximal value of primal iterates.	1e+20
dual_inf_tol	Desired threshold for the dual infeasibility.	1
max_cpu_time	Maximum number of CPU seconds.	1e+20
max_iter	Maximum number of iterations.	GAMS iterlim
max_wall_time	Maximum number of walltime clock seconds.	GAMS reslim
mu_target	Desired value of complementarity.	0
tol	Desired convergence tolerance (relative).	1e-08
Options for expert users
s_max	Scaling threshold for the NLP error.	100

Output

Option	Description	Default
inf_pr_output	Determines what value is printed in the "inf_pr" output column.	original
print_eval_error	Switch to enable printing information about function evaluation errors into the GAMS listing file.	yes
print_frequency_iter	Determines at which iteration frequency the summarizing iteration output line should be printed.	1
print_frequency_time	Determines at which time frequency the summarizing iteration output line should be printed.	0
print_info_string	Enables printing of additional info string at end of iteration output.	no
print_level	Output verbosity level.	5
print_options_mode	format in which to print options documentation	text
print_timing_statistics	Switch to print timing statistics.	no
report_mininfeas_solution	Switch to report intermediate solution with minimal constraint violation to GAMS if the final solution is not feasible.	no
Options for expert users
print_advanced_options	whether to print also advanced options	no

NLP

Option	Description	Default
bound_relax_factor	Factor for initial relaxation of the bounds.	1e-10
check_derivatives_for_naninf	Indicates whether it is desired to check for Nan/Inf in derivative matrices	no
fixed_variable_treatment	Determines how fixed variables should be handled.	make_parameter
honor_original_bounds	Indicates whether final points should be projected into original bounds.	no
Options for expert users
dependency_detection_with_rhs	Indicates if the right hand sides of the constraints should be considered in addition to gradients during dependency detection	no
dependency_detector	Indicates which linear solver should be used to detect linearly dependent equality constraints.	none
kappa_d	Weight for linear damping term (to handle one-sided bounds).	1e-05

NLP Scaling

Option	Description	Default
nlp_scaling_max_gradient	Maximum gradient after NLP scaling.	100
nlp_scaling_method	Select the technique used for scaling the NLP.	gradient-based if GAMS scaleopt is not set, otherwise none
nlp_scaling_min_value	Minimum value of gradient-based scaling values.	1e-08
Options for expert users
nlp_scaling_constr_target_gradient	Target value for constraint function gradient size.	0
nlp_scaling_obj_target_gradient	Target value for objective function gradient size.	0

Initialization

Option	Description	Default
bound_frac	Desired minimum relative distance from the initial point to bound.	0.01
bound_mult_init_method	Initialization method for bound multipliers	constant
bound_mult_init_val	Initial value for the bound multipliers.	1
bound_push	Desired minimum absolute distance from the initial point to bound.	0.01
constr_mult_init_max	Maximum allowed least-square guess of constraint multipliers.	1000
least_square_init_duals	Least square initialization of all dual variables	no
least_square_init_primal	Least square initialization of the primal variables	no
slack_bound_frac	Desired minimum relative distance from the initial slack to bound.	0.01
slack_bound_push	Desired minimum absolute distance from the initial slack to bound.	0.01

Warm Start

Option	Description	Default
warm_start_bound_frac	same as bound_frac for the regular initializer	0.001
warm_start_bound_push	same as bound_push for the regular initializer	0.001
warm_start_init_point	Warm-start for initial point	yes, if run on modified model instance (e.g., from GUSS), otherwise no
warm_start_mult_bound_push	same as mult_bound_push for the regular initializer	0.001
warm_start_mult_init_max	Maximum initial value for the equality multipliers.	1e+06
warm_start_slack_bound_frac	same as slack_bound_frac for the regular initializer	0.001
warm_start_slack_bound_push	same as slack_bound_push for the regular initializer	0.001
Options for expert users
warm_start_target_mu		0

Miscellaneous

Option	Description	Default
timing_statistics	Indicates whether to measure time spend in components of Ipopt and NLP evaluation	no
Options for expert users
replace_bounds	Whether all variable bounds should be replaced by inequality constraints	no

Barrier Parameter Update

Option	Description	Default
adaptive_mu_globalization	Globalization strategy for the adaptive mu selection mode.	obj-constr-filter
barrier_tol_factor	Factor for mu in barrier stop test.	10
fixed_mu_oracle	Oracle for the barrier parameter when switching to fixed mode.	average_compl
mu_init	Initial value for the barrier parameter.	0.1
mu_linear_decrease_factor	Determines linear decrease rate of barrier parameter.	0.2
mu_max	Maximum value for barrier parameter.	100000
mu_max_fact	Factor for initialization of maximum value for barrier parameter.	1000
mu_min	Minimum value for barrier parameter.	1e-11
mu_oracle	Oracle for a new barrier parameter in the adaptive strategy.	quality-function
mu_strategy	Update strategy for barrier parameter.	adaptive
mu_superlinear_decrease_power	Determines superlinear decrease rate of barrier parameter.	1.5
quality_function_max_section_steps	Maximum number of search steps during direct search procedure determining the optimal centering parameter.	8
Options for expert users
adaptive_mu_kkt_norm_type	Norm used for the KKT error in the adaptive mu globalization strategies.	2-norm-squared
adaptive_mu_kkterror_red_fact	Sufficient decrease factor for "kkt-error" globalization strategy.	0.9999
adaptive_mu_kkterror_red_iters	Maximum number of iterations requiring sufficient progress.	4
adaptive_mu_monotone_init_factor	Determines the initial value of the barrier parameter when switching to the monotone mode.	0.8
adaptive_mu_restore_previous_iterate	Indicates if the previous accepted iterate should be restored if the monotone mode is entered.	no
filter_margin_fact	Factor determining width of margin for obj-constr-filter adaptive globalization strategy.	1e-05
filter_max_margin	Maximum width of margin in obj-constr-filter adaptive globalization strategy.	1
mu_allow_fast_monotone_decrease	Allow skipping of barrier problem if barrier test is already met.	yes
quality_function_balancing_term	The balancing term included in the quality function for centrality.	none
quality_function_centrality	The penalty term for centrality that is included in quality function.	none
quality_function_norm_type	Norm used for components of the quality function.	2-norm-squared
quality_function_section_qf_tol	Tolerance for the golden section search procedure determining the optimal centering parameter (in the function value space).	0
quality_function_section_sigma_tol	Tolerance for the section search procedure determining the optimal centering parameter (in sigma space).	0.01
sigma_max	Maximum value of the centering parameter.	100
sigma_min	Minimum value of the centering parameter.	1e-06
tau_min	Lower bound on fraction-to-the-boundary parameter tau.	0.99

Line Search

Option	Description	Default
accept_every_trial_step	Always accept the first trial step.	no
alpha_for_y	Method to determine the step size for constraint multipliers (alpha_y) .	primal
alpha_for_y_tol	Tolerance for switching to full equality multiplier steps.	10
max_soc	Maximum number of second order correction trial steps at each iteration.	4
recalc_y	Tells the algorithm to recalculate the equality and inequality multipliers as least square estimates.	no
recalc_y_feas_tol	Feasibility threshold for recomputation of multipliers.	1e-06
soc_method	Ways to apply second order correction	0
watchdog_shortened_iter_trigger	Number of shortened iterations that trigger the watchdog.	10
watchdog_trial_iter_max	Maximum number of watchdog iterations.	3
Options for expert users
accept_after_max_steps	Accept a trial point after maximal this number of steps even if it does not satisfy line search conditions.	-1
alpha_min_frac	Safety factor for the minimal step size (before switching to restoration phase).	0.05
alpha_red_factor	Fractional reduction of the trial step size in the backtracking line search.	0.5
constraint_violation_norm_type	Norm to be used for the constraint violation in the line search.	1-norm
corrector_compl_avrg_red_fact	Complementarity tolerance factor for accepting corrector step.	1
corrector_type	The type of corrector steps that should be taken.	none
delta	Multiplier for constraint violation in the switching rule.	1
eta_phi	Relaxation factor in the Armijo condition.	1e-08
filter_reset_trigger	Number of iterations that trigger the filter reset.	5
gamma_phi	Relaxation factor in the filter margin for the barrier function.	1e-08
gamma_theta	Relaxation factor in the filter margin for the constraint violation.	1e-05
kappa_sigma	Factor limiting the deviation of dual variables from primal estimates.	1e+10
kappa_soc	Factor in the sufficient reduction rule for second order correction.	0.99
line_search_method	Globalization method used in backtracking line search	filter
max_filter_resets	Maximal allowed number of filter resets	5
nu_inc	Increment of the penalty parameter.	0.0001
nu_init	Initial value of the penalty parameter.	1e-06
obj_max_inc	Determines the upper bound on the acceptable increase of barrier objective function.	5
rho	Value in penalty parameter update formula.	0.1
s_phi	Exponent for linear barrier function model in the switching rule.	2.3
s_theta	Exponent for current constraint violation in the switching rule.	1.1
skip_corr_if_neg_curv	Whether to skip the corrector step in negative curvature iteration.	yes
skip_corr_in_monotone_mode	Whether to skip the corrector step during monotone barrier parameter mode.	yes
slack_move	Correction size for very small slacks.	1.81899e-12
theta_max_fact	Determines upper bound for constraint violation in the filter.	10000
theta_min_fact	Determines constraint violation threshold in the switching rule.	0.0001
tiny_step_tol	Tolerance for detecting numerically insignificant steps.	2.22045e-15
tiny_step_y_tol	Tolerance for quitting because of numerically insignificant steps.	0.01

Linear Solver

Option	Description	Default
hsllib	Name of library containing HSL routines for load at runtime	libhsl.so (Linux), libhsl.dylib (macOS), libhsl.dll (Windows)
linear_scaling_on_demand	Flag indicating that linear scaling is only done if it seems required.	yes
linear_solver	Linear solver used for step computations.	ma27, if IpoptH, otherwise mumps
linear_system_scaling	Method for scaling the linear system.	mc19, if IpoptH, otherwise none
pardisolib	Name of library containing Pardiso routines (from pardiso-project.org) for load at runtime	libpardiso.so (Linux), libpardiso.dylib (macOS), libpardiso.dll (Windows)

Step Calculation

Option	Description	Default
fast_step_computation	Indicates if the linear system should be solved quickly.	no
first_hessian_perturbation	Size of first x-s perturbation tried.	0.0001
jacobian_regularization_value	Size of the regularization for rank-deficient constraint Jacobians.	1e-08
max_hessian_perturbation	Maximum value of regularization parameter for handling negative curvature.	1e+20
max_refinement_steps	Maximum number of iterative refinement steps per linear system solve.	10
mehrotra_algorithm	Indicates whether to do Mehrotra's predictor-corrector algorithm.	no
min_hessian_perturbation	Smallest perturbation of the Hessian block.	1e-20
min_refinement_steps	Minimum number of iterative refinement steps per linear system solve.	1
neg_curv_test_reg	Whether to do the curvature test with the primal regularization (see Zavala and Chiang, 2014).	yes
neg_curv_test_tol	Tolerance for heuristic to ignore wrong inertia.	0
perturb_dec_fact	Decrease factor for x-s perturbation.	0.333333
perturb_inc_fact	Increase factor for x-s perturbation.	8
perturb_inc_fact_first	Increase factor for x-s perturbation for very first perturbation.	100
Options for expert users
jacobian_regularization_exponent	Exponent for mu in the regularization for rank-deficient constraint Jacobians.	0.25
perturb_always_cd	Active permanent perturbation of constraint linearization.	no
residual_improvement_factor	Minimal required reduction of residual test ratio in iterative refinement.	1
residual_ratio_max	Iterative refinement tolerance	1e-10
residual_ratio_singular	Threshold for declaring linear system singular after failed iterative refinement.	1e-05

Restoration Phase

Option	Description	Default
bound_mult_reset_threshold	Threshold for resetting bound multipliers after the restoration phase.	1000
constr_mult_reset_threshold	Threshold for resetting equality and inequality multipliers after restoration phase.	0
evaluate_orig_obj_at_resto_trial	Determines if the original objective function should be evaluated at restoration phase trial points.	yes
expect_infeasible_problem	Enable heuristics to quickly detect an infeasible problem.	no
expect_infeasible_problem_ctol	Threshold for disabling "expect_infeasible_problem" option.	0.001
expect_infeasible_problem_ytol	Multiplier threshold for activating "expect_infeasible_problem" option.	1e+08
required_infeasibility_reduction	Required reduction of infeasibility before leaving restoration phase.	0.9
soft_resto_pderror_reduction_factor	Required reduction in primal-dual error in the soft restoration phase.	0.9999
start_with_resto	Whether to switch to restoration phase in first iteration.	no
Options for expert users
max_resto_iter	Maximum number of successive iterations in restoration phase.	3000000
max_soft_resto_iters	Maximum number of iterations performed successively in soft restoration phase.	10
resto_failure_feasibility_threshold	Threshold for primal infeasibility to declare failure of restoration phase.	0
resto_penalty_parameter	Penalty parameter in the restoration phase objective function.	1000
resto_proximity_weight	Weighting factor for the proximity term in restoration phase objective.	1

Hessian Approximation

Option	Description	Default
hessian_approximation	Indicates what Hessian information is to be used.	exact
limited_memory_init_val	Value for B0 in low-rank update.	1
limited_memory_init_val_max	Upper bound on value for B0 in low-rank update.	1e+08
limited_memory_init_val_min	Lower bound on value for B0 in low-rank update.	1e-08
limited_memory_initialization	Initialization strategy for the limited memory quasi-Newton approximation.	scalar1
limited_memory_max_history	Maximum size of the history for the limited quasi-Newton Hessian approximation.	6
limited_memory_max_skipping	Threshold for successive iterations where update is skipped.	2
limited_memory_special_for_resto	Determines if the quasi-Newton updates should be special during the restoration phase.	no
limited_memory_update_type	Quasi-Newton update formula for the limited memory quasi-Newton approximation.	bfgs
Options for expert users
hessian_approximation_space	Indicates in which subspace the Hessian information is to be approximated.	nonlinear-variables
limited_memory_aug_solver	Strategy for solving the augmented system for low-rank Hessian.	sherman-morrison

MA27 Linear Solver

Option	Description	Default
ma27_la_init_factor	Real workspace memory for MA27.	5
ma27_liw_init_factor	Integer workspace memory for MA27.	5
ma27_meminc_factor	Increment factor for workspace size for MA27.	2
ma27_pivtol	Pivot tolerance for the linear solver MA27.	1e-08
ma27_pivtolmax	Maximum pivot tolerance for the linear solver MA27.	0.0001
ma27_print_level	Debug printing level for the linear solver MA27	0
Options for expert users
ma27_ignore_singularity	Whether to use MA27's ability to solve a linear system even if the matrix is singular.	no
ma27_skip_inertia_check	Whether to always pretend that inertia is correct.	no

MA57 Linear Solver

Option	Description	Default
ma57_automatic_scaling	Controls whether to enable automatic scaling in MA57	no
ma57_block_size	Controls block size used by Level 3 BLAS in MA57BD	16
ma57_node_amalgamation	Node amalgamation parameter	16
ma57_pivot_order	Controls pivot order in MA57	5
ma57_pivtol	Pivot tolerance for the linear solver MA57.	1e-08
ma57_pivtolmax	Maximum pivot tolerance for the linear solver MA57.	0.0001
ma57_pre_alloc	Safety factor for work space memory allocation for the linear solver MA57.	1.05
ma57_print_level	Debug printing level for the linear solver MA57	0
ma57_small_pivot_flag	Handling of small pivots	0

MA77 Linear Solver

Option	Description	Default
ma77_buffer_lpage	Number of scalars per MA77 in-core buffer page in the out-of-core solver MA77	4096
ma77_buffer_npage	Number of pages that make up MA77 buffer	1600
ma77_file_size	Target size of each temporary file for MA77, scalars per type	2097152
ma77_maxstore	Maximum storage size for MA77 in-core mode	0
ma77_nemin	Node Amalgamation parameter	8
ma77_order	Controls type of ordering used by MA77	metis
ma77_print_level	Debug printing level for the linear solver MA77	-1
ma77_small	Zero Pivot Threshold	1e-20
ma77_static	Static Pivoting Threshold	0
ma77_u	Pivoting Threshold	1e-08
ma77_umax	Maximum Pivoting Threshold	0.0001

MA86 Linear Solver

Option	Description	Default
ma86_nemin	Node Amalgamation parameter	32
ma86_order	Controls type of ordering	auto
ma86_print_level	Debug printing level	-1
ma86_scaling	Controls scaling of matrix	mc64
ma86_small	Zero Pivot Threshold	1e-20
ma86_static	Static Pivoting Threshold	0
ma86_u	Pivoting Threshold	1e-08
ma86_umax	Maximum Pivoting Threshold	0.0001

MA97 Linear Solver

Option	Description	Default
ma97_nemin	Node Amalgamation parameter	8
ma97_order	Controls type of ordering	auto
ma97_print_level	Debug printing level	-1
ma97_scaling	Specifies strategy for scaling	dynamic
ma97_small	Zero Pivot Threshold	1e-20
ma97_u	Pivoting Threshold	1e-08
ma97_umax	Maximum Pivoting Threshold	0.0001
Options for expert users
ma97_scaling1	First scaling.	mc64
ma97_scaling2	Second scaling.	mc64
ma97_scaling3	Third scaling.	mc64
ma97_solve_blas3	Controls if blas2 or blas3 routines are used for solve	no
ma97_switch1	First switch, determine when ma97_scaling1 is enabled.	od_hd_reuse
ma97_switch2	Second switch, determine when ma97_scaling2 is enabled.	never
ma97_switch3	Third switch, determine when ma97_scaling3 is enabled.	never

Pardiso (pardiso-project.org) Linear Solver

Option	Description	Default
pardiso_matching_strategy	Matching strategy to be used by Pardiso	complete+2x2
pardiso_max_iterative_refinement_steps	Limit on number of iterative refinement steps.	0
pardiso_msglvl	Pardiso message level	0
pardiso_order	Controls the fill-in reduction ordering algorithm for the input matrix.	metis
Options for expert users
pardiso_iter_coarse_size	Maximum Size of Coarse Grid Matrix	5000
pardiso_iter_dropping_factor	dropping value for incomplete factor	0.5
pardiso_iter_dropping_schur	dropping value for sparsify schur complement factor	0.1
pardiso_iter_inverse_norm_factor		5e+06
pardiso_iter_max_levels	Maximum Size of Grid Levels	10
pardiso_iter_max_row_fill	max fill for each row	10000000
pardiso_iter_relative_tol	Relative Residual Convergence	1e-06
pardiso_iterative	Switch for iterative solver in Pardiso library	no
pardiso_max_droptol_corrections	Maximal number of decreases of drop tolerance during one solve.	4
pardiso_max_iter	Maximum number of Krylov-Subspace Iteration	500
pardiso_redo_symbolic_fact_only_if_inertia_wrong	Toggle for handling case when elements were perturbed by Pardiso.	no
pardiso_repeated_perturbation_means_singular	Whether to assume that matrix is singular if elements were perturbed after recent symbolic factorization.	no
pardiso_skip_inertia_check	Whether to pretend that inertia is correct.	no

Pardiso (MKL) Linear Solver

Option	Description	Default
pardisomkl_matching_strategy	Matching strategy to be used by Pardiso	complete+2x2
pardisomkl_max_iterative_refinement_steps	Limit on number of iterative refinement steps.	1
pardisomkl_msglvl	Pardiso message level	0
pardisomkl_order	Controls the fill-in reduction ordering algorithm for the input matrix.	metis
Options for expert users
pardisomkl_redo_symbolic_fact_only_if_inertia_wrong	Toggle for handling case when elements were perturbed by Pardiso.	no
pardisomkl_repeated_perturbation_means_singular	Whether to assume that matrix is singular if elements were perturbed after recent symbolic factorization.	no
pardisomkl_skip_inertia_check	Whether to pretend that inertia is correct.	no

Mumps Linear Solver

Option	Description	Default
mumps_mem_percent	Percentage increase in the estimated working space for MUMPS.	1000
mumps_permuting_scaling	Controls permuting and scaling in MUMPS	7
mumps_pivot_order	Controls pivot order in MUMPS	7
mumps_pivtol	Pivot tolerance for the linear solver MUMPS.	1e-06
mumps_pivtolmax	Maximum pivot tolerance for the linear solver MUMPS.	0.1
mumps_print_level	Debug printing level for the linear solver MUMPS	0
mumps_scaling	Controls scaling in MUMPS	77
Options for expert users
mumps_dep_tol	Threshold to consider a pivot at zero in detection of linearly dependent constraints with MUMPS.	0

Detailed Options Description

accept_after_max_steps (advanced): Accept a trial point after maximal this number of steps even if it does not satisfy line search conditions. ↵

Setting this to -1 disables this option.

Range: {-1, ..., ∞}

Default: -1

accept_every_trial_step: Always accept the first trial step. ↵

Setting this option to "yes" essentially disables the line search and makes the algorithm take aggressive steps, without global convergence guarantees.

Range: yes, no

Default: no

acceptable_compl_inf_tol: "Acceptance" threshold for the complementarity conditions. ↵

Absolute tolerance on the complementarity. "Acceptable" termination requires that the max-norm of the (unscaled) complementarity is less than this threshold; see also acceptable_tol.

Range: (0, ∞]

Default: 0.01

acceptable_constr_viol_tol: "Acceptance" threshold for the constraint violation. ↵

Absolute tolerance on the constraint violation. "Acceptable" termination requires that the max-norm of the (unscaled) constraint violation is less than this threshold; see also acceptable_tol.

Range: (0, ∞]

Default: 0.01

acceptable_dual_inf_tol: "Acceptance" threshold for the dual infeasibility. ↵

Absolute tolerance on the dual infeasibility. "Acceptable" termination requires that the (max-norm of the unscaled) dual infeasibility is less than this threshold; see also acceptable_tol.

Range: (0, ∞]

Default: 1e+10

acceptable_iter: Number of "acceptable" iterates before triggering termination. ↵

If the algorithm encounters this many successive "acceptable" iterates (see "acceptable_tol"), it terminates, assuming that the problem has been solved to best possible accuracy given round-off. If it is set to zero, this heuristic is disabled.

Range: {0, ..., ∞}

Default: 0

acceptable_obj_change_tol: "Acceptance" stopping criterion based on objective function change. ↵

If the relative change of the objective function (scaled by Max(1,|f(x)|)) is less than this value, this part of the acceptable tolerance termination is satisfied; see also acceptable_tol. This is useful for the quasi-Newton option, which has trouble to bring down the dual infeasibility.

Range: [0, ∞]

Default: 1e+20

acceptable_tol: "Acceptable" convergence tolerance (relative). ↵

Determines which (scaled) overall optimality error is considered to be "acceptable". There are two levels of termination criteria. If the usual "desired" tolerances (see tol, dual_inf_tol etc) are satisfied at an iteration, the algorithm immediately terminates with a success message. On the other hand, if the algorithm encounters "acceptable_iter" many iterations in a row that are considered "acceptable", it will terminate before the desired convergence tolerance is met. This is useful in cases where the algorithm might not be able to achieve the "desired" level of accuracy.

Range: (0, ∞]

Default: 1e-06

adaptive_mu_globalization: Globalization strategy for the adaptive mu selection mode. ↵

To achieve global convergence of the adaptive version, the algorithm has to switch to the monotone mode (Fiacco-McCormick approach) when convergence does not seem to appear. This option sets the criterion used to decide when to do this switch. (Only used if option "mu_strategy" is chosen as "adaptive".)

value	meaning
kkt-error	nonmonotone decrease of kkt-error
obj-constr-filter	2-dim filter for objective and constraint violation
never-monotone-mode	disables globalization

Default: obj-constr-filter

adaptive_mu_kkt_norm_type (advanced): Norm used for the KKT error in the adaptive mu globalization strategies. ↵

When computing the KKT error for the globalization strategies, the norm to be used is specified with this option. Note, this option is also used in the QualityFunctionMuOracle.

value	meaning
1-norm	use the 1-norm (abs sum)
2-norm-squared	use the 2-norm squared (sum of squares)
max-norm	use the infinity norm (max)
2-norm	use 2-norm

Default: 2-norm-squared

adaptive_mu_kkterror_red_fact (advanced): Sufficient decrease factor for "kkt-error" globalization strategy. ↵

For the "kkt-error" based globalization strategy, the error must decrease by this factor to be deemed sufficient decrease.

Range: (0, 1)

Default: 0.9999

adaptive_mu_kkterror_red_iters (advanced): Maximum number of iterations requiring sufficient progress. ↵

For the "kkt-error" based globalization strategy, sufficient progress must be made for "adaptive_mu_kkterror_red_iters" iterations. If this number of iterations is exceeded, the globalization strategy switches to the monotone mode.

Range: {0, ..., ∞}

Default: 4

adaptive_mu_monotone_init_factor (advanced): Determines the initial value of the barrier parameter when switching to the monotone mode. ↵

When the globalization strategy for the adaptive barrier algorithm switches to the monotone mode and fixed_mu_oracle is chosen as "average_compl", the barrier parameter is set to the current average complementarity times the value of "adaptive_mu_monotone_init_factor".

Range: (0, ∞]

Default: 0.8

adaptive_mu_restore_previous_iterate (advanced): Indicates if the previous accepted iterate should be restored if the monotone mode is entered. ↵

When the globalization strategy for the adaptive barrier algorithm switches to the monotone mode, it can either start from the most recent iterate (no), or from the last iterate that was accepted (yes).

Range: yes, no

Default: no

alpha_for_y: Method to determine the step size for constraint multipliers (alpha_y) . ↵

value	meaning
primal	use primal step size
bound-mult	use step size for the bound multipliers (good for LPs)
min	use the min of primal and bound multipliers
max	use the max of primal and bound multipliers
full	take a full step of size one
min-dual-infeas	choose step size minimizing new dual infeasibility
safer-min-dual-infeas	like "min_dual_infeas", but safeguarded by "min" and "max"
primal-and-full	use the primal step size, and full step if delta_x ≤ alpha_for_y_tol
dual-and-full	use the dual step size, and full step if delta_x ≤ alpha_for_y_tol
acceptor	Call LSAcceptor to get step size for y

Default: primal

alpha_for_y_tol: Tolerance for switching to full equality multiplier steps. ↵

This is only relevant if "alpha_for_y" is chosen "primal-and-full" or "dual-and-full". The step size for the equality constraint multipliers is taken to be one if the max-norm of the primal step is less than this tolerance.

Range: [0, ∞]

Default: 10

alpha_min_frac (advanced): Safety factor for the minimal step size (before switching to restoration phase). ↵

This is gamma_alpha in Eqn. (23) in the implementation paper.

Range: (0, 1)

Default: 0.05

alpha_red_factor (advanced): Fractional reduction of the trial step size in the backtracking line search. ↵

At every step of the backtracking line search, the trial step size is reduced by this factor.

Range: (0, 1)

Default: 0.5

barrier_tol_factor: Factor for mu in barrier stop test. ↵

The convergence tolerance for each barrier problem in the monotone mode is the value of the barrier parameter times "barrier_tol_factor". This option is also used in the adaptive mu strategy during the monotone mode. This is kappa_epsilon in implementation paper.

Range: (0, ∞]

Default: 10

bound_frac: Desired minimum relative distance from the initial point to bound. ↵

Determines how much the initial point might have to be modified in order to be sufficiently inside the bounds (together with "bound_push"). (This is kappa_2 in Section 3.6 of implementation paper.)

Range: (0, 0.5]

Default: 0.01

bound_mult_init_method: Initialization method for bound multipliers ↵

This option defines how the iterates for the bound multipliers are initialized. If "constant" is chosen, then all bound multipliers are initialized to the value of "bound_mult_init_val". If "mu-based" is chosen, then each value is initialized to the the value of "mu_init" divided by the corresponding slack variable. This latter option might be useful if the starting point is close to the optimal solution.

value	meaning
constant	set all bound multipliers to the value of bound_mult_init_val
mu-based	initialize to mu_init/x_slack

Default: constant

bound_mult_init_val: Initial value for the bound multipliers. ↵

All dual variables corresponding to bound constraints are initialized to this value.

Range: (0, ∞]

Default: 1

bound_mult_reset_threshold: Threshold for resetting bound multipliers after the restoration phase. ↵

After returning from the restoration phase, the bound multipliers are updated with a Newton step for complementarity. Here, the change in the primal variables during the entire restoration phase is taken to be the corresponding primal Newton step. However, if after the update the largest bound multiplier exceeds the threshold specified by this option, the multipliers are all reset to 1.

Range: [0, ∞]

Default: 1000

bound_push: Desired minimum absolute distance from the initial point to bound. ↵

Determines how much the initial point might have to be modified in order to be sufficiently inside the bounds (together with "bound_frac"). (This is kappa_1 in Section 3.6 of implementation paper.)

Range: (0, ∞]

Default: 0.01

bound_relax_factor: Factor for initial relaxation of the bounds. ↵

Before start of the optimization, the bounds given by the user are relaxed. This option sets the factor for this relaxation. Additional, the constraint violation tolerance constr_viol_tol is used to bound the relaxation by an absolute value. If it is set to zero, then then bounds relaxation is disabled. See Eqn.(35) in implementation paper. Note that the constraint violation reported by Ipopt at the end of the solution process does not include violations of the original (non-relaxed) variable bounds. See also option honor_original_bounds.

Range: [0, ∞]

Default: 1e-10

check_derivatives_for_naninf: Indicates whether it is desired to check for Nan/Inf in derivative matrices ↵

Activating this option will cause an error if an invalid number is detected in the constraint Jacobians or the Lagrangian Hessian. If this is not activated, the test is skipped, and the algorithm might proceed with invalid numbers and fail. If test is activated and an invalid number is detected, the matrix is written to output with print_level corresponding to J_MOREDETAILED (7); so beware of large output!

Range: yes, no

Default: no

compl_inf_tol: Desired threshold for the complementarity conditions. ↵

Absolute tolerance on the complementarity. Successful termination requires that the max-norm of the (unscaled) complementarity is less than this threshold.

Range: (0, ∞]

Default: 0.0001

constr_mult_init_max: Maximum allowed least-square guess of constraint multipliers. ↵

Determines how large the initial least-square guesses of the constraint multipliers are allowed to be (in max-norm). If the guess is larger than this value, it is discarded and all constraint multipliers are set to zero. This options is also used when initializing the restoration phase. By default, "resto.constr_mult_init_max" (the one used in RestoIterateInitializer) is set to zero.

Range: [0, ∞]

Default: 1000

constr_mult_reset_threshold: Threshold for resetting equality and inequality multipliers after restoration phase. ↵

After returning from the restoration phase, the constraint multipliers are recomputed by a least square estimate. This option triggers when those least-square estimates should be ignored.

Range: [0, ∞]

Default: 0

constr_viol_tol: Desired threshold for the constraint and variable bound violation. ↵

Absolute tolerance on the constraint and variable bound violation. Successful termination requires that the max-norm of the (unscaled) constraint violation is less than this threshold. If option bound_relax_factor is not zero 0, then Ipopt relaxes given variable bounds. The value of constr_viol_tol is used to restrict the absolute amount of this bound relaxation.

Range: (0, ∞]

Default: 1e-06

constraint_violation_norm_type (advanced): Norm to be used for the constraint violation in the line search. ↵

Determines which norm should be used when the algorithm computes the constraint violation in the line search.

value	meaning
1-norm	use the 1-norm
2-norm	use the 2-norm
max-norm	use the infinity norm

Default: 1-norm

corrector_compl_avrg_red_fact (advanced): Complementarity tolerance factor for accepting corrector step. ↵

This option determines the factor by which complementarity is allowed to increase for a corrector step to be accepted. Changing this option is experimental.

Range: (0, ∞]

Default: 1

corrector_type (advanced): The type of corrector steps that should be taken. ↵

If "mu_strategy" is "adaptive", this option determines what kind of corrector steps should be tried. Changing this option is experimental.

value	meaning
none	no corrector
affine	corrector step towards mu=0
primal-dual	corrector step towards current mu

Default: none

delta (advanced): Multiplier for constraint violation in the switching rule. ↵

See Eqn. (19) in the implementation paper.

Range: (0, ∞]

Default: 1

dependency_detection_with_rhs (advanced): Indicates if the right hand sides of the constraints should be considered in addition to gradients during dependency detection ↵

Range: yes, no

Default: no

dependency_detector (advanced): Indicates which linear solver should be used to detect linearly dependent equality constraints. ↵

This is experimental and does not work well.

value	meaning
none	don't check; no extra work at beginning
mumps	use MUMPS

Default: none

diverging_iterates_tol: Threshold for maximal value of primal iterates. ↵

If any component of the primal iterates exceeded this value (in absolute terms), the optimization is aborted with the exit message that the iterates seem to be diverging.

Range: (0, ∞]

Default: 1e+20

dual_inf_tol: Desired threshold for the dual infeasibility. ↵

Absolute tolerance on the dual infeasibility. Successful termination requires that the max-norm of the (unscaled) dual infeasibility is less than this threshold.

Range: (0, ∞]

Default: 1

eta_phi (advanced): Relaxation factor in the Armijo condition. ↵

See Eqn. (20) in the implementation paper.

Range: (0, 0.5)

Default: 1e-08

evaluate_orig_obj_at_resto_trial: Determines if the original objective function should be evaluated at restoration phase trial points. ↵

Enabling this option makes the restoration phase algorithm evaluate the objective function of the original problem at every trial point encountered during the restoration phase, even if this value is not required. In this way, it is guaranteed that the original objective function can be evaluated without error at all accepted iterates; otherwise the algorithm might fail at a point where the restoration phase accepts an iterate that is good for the restoration phase problem, but not the original problem. On the other hand, if the evaluation of the original objective is expensive, this might be costly.

Range: yes, no

Default: yes

expect_infeasible_problem: Enable heuristics to quickly detect an infeasible problem. ↵

This options is meant to activate heuristics that may speed up the infeasibility determination if you expect that there is a good chance for the problem to be infeasible. In the filter line search procedure, the restoration phase is called more quickly than usually, and more reduction in the constraint violation is enforced before the restoration phase is left. If the problem is square, this option is enabled automatically.

Range: yes, no

Default: no

expect_infeasible_problem_ctol: Threshold for disabling "expect_infeasible_problem" option. ↵

If the constraint violation becomes smaller than this threshold, the "expect_infeasible_problem" heuristics in the filter line search are disabled. If the problem is square, this options is set to 0.

Range: [0, ∞]

Default: 0.001

expect_infeasible_problem_ytol: Multiplier threshold for activating "expect_infeasible_problem" option. ↵

If the max norm of the constraint multipliers becomes larger than this value and "expect_infeasible_problem" is chosen, then the restoration phase is entered.

Range: (0, ∞]

Default: 1e+08

fast_step_computation: Indicates if the linear system should be solved quickly. ↵

If enabled, the algorithm assumes that the linear system that is solved to obtain the search direction is solved sufficiently well. In that case, no residuals are computed to verify the solution and the computation of the search direction is a little faster.

Range: yes, no

Default: no

filter_margin_fact (advanced): Factor determining width of margin for obj-constr-filter adaptive globalization strategy. ↵

When using the adaptive globalization strategy, "obj-constr-filter", sufficient progress for a filter entry is defined as follows: (new obj) < (filter obj) - filter_margin_fact*(new constr-viol) OR (new constr-viol) < (filter constr-viol) - filter_margin_fact*(new constr-viol). For the description of the "kkt-error-filter" option see "filter_max_margin".

Range: (0, 1)

Default: 1e-05

filter_max_margin (advanced): Maximum width of margin in obj-constr-filter adaptive globalization strategy. ↵

Range: (0, ∞]

Default: 1

filter_reset_trigger (advanced): Number of iterations that trigger the filter reset. ↵

If the filter reset heuristic is active and the number of successive iterations in which the last rejected trial step size was rejected because of the filter, the filter is reset.

Range: {1, ..., ∞}

Default: 5

first_hessian_perturbation: Size of first x-s perturbation tried. ↵

The first value tried for the x-s perturbation in the inertia correction scheme. This is delta_0 in the implementation paper.

Range: (0, ∞]

Default: 0.0001

fixed_mu_oracle: Oracle for the barrier parameter when switching to fixed mode. ↵

Determines how the first value of the barrier parameter should be computed when switching to the "monotone mode" in the adaptive strategy. (Only considered if "adaptive" is selected for option "mu_strategy".)

value	meaning
probing	Mehrotra's probing heuristic
loqo	LOQO's centrality rule
quality-function	minimize a quality function
average_compl	base on current average complementarity

Default: average_compl

fixed_variable_treatment: Determines how fixed variables should be handled. ↵

The main difference between those options is that the starting point in the "make_constraint" case still has the fixed variables at their given values, whereas in the case "make_parameter(_nodual)" the functions are always evaluated with the fixed values for those variables. Also, for "relax_bounds", the fixing bound constraints are relaxed (according to" bound_relax_factor"). For all but "make_parameter_nodual", bound multipliers are computed for the fixed variables.

value	meaning
make_parameter	Remove fixed variable from optimization variables
make_parameter_nodual	Remove fixed variable from optimization variables and do not compute bound multipliers for fixed variables
make_constraint	Add equality constraints fixing variables
relax_bounds	Relax fixing bound constraints

Default: make_parameter

gamma_phi (advanced): Relaxation factor in the filter margin for the barrier function. ↵

See Eqn. (18a) in the implementation paper.

Range: (0, 1)

Default: 1e-08

gamma_theta (advanced): Relaxation factor in the filter margin for the constraint violation. ↵

See Eqn. (18b) in the implementation paper.

Range: (0, 1)

Default: 1e-05

hessian_approximation: Indicates what Hessian information is to be used. ↵

This determines which kind of information for the Hessian of the Lagrangian function is used by the algorithm.

value	meaning
exact	Use second derivatives provided by the NLP.
limited-memory	Perform a limited-memory quasi-Newton approximation

Default: exact

hessian_approximation_space (advanced): Indicates in which subspace the Hessian information is to be approximated. ↵

value	meaning
nonlinear-variables	only in space of nonlinear variables.
all-variables	in space of all variables (without slacks)

Default: nonlinear-variables

honor_original_bounds: Indicates whether final points should be projected into original bounds. ↵

Ipopt might relax the bounds during the optimization (see, e.g., option "bound_relax_factor"). This option determines whether the final point should be projected back into the user-provide original bounds after the optimization. Note that violations of constraints and complementarity reported by Ipopt at the end of the solution process are for the non-projected point.

Range: yes, no

Default: no

hsllib: Name of library containing HSL routines for load at runtime ↵

Range: string

Default: libhsl.so (Linux), libhsl.dylib (macOS), libhsl.dll (Windows)

inf_pr_output: Determines what value is printed in the "inf_pr" output column. ↵

Ipopt works with a reformulation of the original problem, where slacks are introduced and the problem might have been scaled. The choice "internal" prints out the constraint violation of this formulation. With "original" the true constraint violation in the original NLP is printed.

value	meaning
internal	max-norm of violation of internal equality constraints
original	maximal constraint violation in original NLP

Default: original

jacobian_regularization_exponent (advanced): Exponent for mu in the regularization for rank-deficient constraint Jacobians. ↵

This is kappa_c in the implementation paper.

Range: [0, ∞]

Default: 0.25

jacobian_regularization_value: Size of the regularization for rank-deficient constraint Jacobians. ↵

This is bar delta_c in the implementation paper.

Range: [0, ∞]

Default: 1e-08

kappa_d (advanced): Weight for linear damping term (to handle one-sided bounds). ↵

See Section 3.7 in implementation paper.

Range: [0, ∞]

Default: 1e-05

kappa_sigma (advanced): Factor limiting the deviation of dual variables from primal estimates. ↵

If the dual variables deviate from their primal estimates, a correction is performed. See Eqn. (16) in the implementation paper. Setting the value to less than 1 disables the correction.

Range: (0, ∞]

Default: 1e+10

kappa_soc (advanced): Factor in the sufficient reduction rule for second order correction. ↵

This option determines how much a second order correction step must reduce the constraint violation so that further correction steps are attempted. See Step A-5.9 of Algorithm A in the implementation paper.

Range: (0, ∞]

Default: 0.99

least_square_init_duals: Least square initialization of all dual variables ↵

If set to yes, Ipopt tries to compute least-square multipliers (considering ALL dual variables). If successful, the bound multipliers are possibly corrected to be at least bound_mult_init_val. This might be useful if the user doesn't know anything about the starting point, or for solving an LP or QP. This overwrites option "bound_mult_init_method".

value	meaning
no	use bound_mult_init_val and least-square equality constraint multipliers
yes	overwrite user-provided point with least-square estimates

Default: no

least_square_init_primal: Least square initialization of the primal variables ↵

If set to yes, Ipopt ignores the user provided point and solves a least square problem for the primal variables (x and s) to fit the linearized equality and inequality constraints.This might be useful if the user doesn't know anything about the starting point, or for solving an LP or QP.

value	meaning
no	take user-provided point
yes	overwrite user-provided point with least-square estimates

Default: no

limited_memory_aug_solver (advanced): Strategy for solving the augmented system for low-rank Hessian. ↵

value	meaning
sherman-morrison	use Sherman-Morrison formula
extended	use an extended augmented system

Default: sherman-morrison

limited_memory_init_val: Value for B0 in low-rank update. ↵

The starting matrix in the low rank update, B0, is chosen to be this multiple of the identity in the first iteration (when no updates have been performed yet), and is constantly chosen as this value, if "limited_memory_initialization" is "constant".

Range: (0, ∞]

Default: 1

limited_memory_init_val_max: Upper bound on value for B0 in low-rank update. ↵

The starting matrix in the low rank update, B0, is chosen to be this multiple of the identity in the first iteration (when no updates have been performed yet), and is constantly chosen as this value, if "limited_memory_initialization" is "constant".

Range: (0, ∞]

Default: 1e+08

limited_memory_init_val_min: Lower bound on value for B0 in low-rank update. ↵

The starting matrix in the low rank update, B0, is chosen to be this multiple of the identity in the first iteration (when no updates have been performed yet), and is constantly chosen as this value, if "limited_memory_initialization" is "constant".

Range: (0, ∞]

Default: 1e-08

limited_memory_initialization: Initialization strategy for the limited memory quasi-Newton approximation. ↵

Determines how the diagonal Matrix B_0 as the first term in the limited memory approximation should be computed.

value	meaning
scalar1	sigma = s^Ty/s^Ts
scalar2	sigma = y^Ty/s^Ty
scalar3	arithmetic average of scalar1 and scalar2
scalar4	geometric average of scalar1 and scalar2
constant	sigma = limited_memory_init_val

Default: scalar1

limited_memory_max_history: Maximum size of the history for the limited quasi-Newton Hessian approximation. ↵

This option determines the number of most recent iterations that are taken into account for the limited-memory quasi-Newton approximation.

Range: {0, ..., ∞}

Default: 6

limited_memory_max_skipping: Threshold for successive iterations where update is skipped. ↵

If the update is skipped more than this number of successive iterations, the quasi-Newton approximation is reset.

Range: {1, ..., ∞}

Default: 2

limited_memory_special_for_resto: Determines if the quasi-Newton updates should be special during the restoration phase. ↵

Until Nov 2010, Ipopt used a special update during the restoration phase, but it turned out that this does not work well. The new default uses the regular update procedure and it improves results. If for some reason you want to get back to the original update, set this option to "yes".

Range: yes, no

Default: no

limited_memory_update_type: Quasi-Newton update formula for the limited memory quasi-Newton approximation. ↵

value	meaning
bfgs	BFGS update (with skipping)
sr1	SR1 (not working well)

Default: bfgs

line_search_method (advanced): Globalization method used in backtracking line search ↵

Only the "filter" choice is officially supported. But sometimes, good results might be obtained with the other choices.

value	meaning
filter	Filter method
cg-penalty	Chen-Goldfarb penalty function
penalty	Standard penalty function

Default: filter

linear_scaling_on_demand: Flag indicating that linear scaling is only done if it seems required. ↵

This option is only important if a linear scaling method (e.g., mc19) is used. If you choose "no", then the scaling factors are computed for every linear system from the start. This can be quite expensive. Choosing "yes" means that the algorithm will start the scaling method only when the solutions to the linear system seem not good, and then use it until the end.

Range: yes, no

Default: yes

linear_solver: Linear solver used for step computations. ↵

Determines which linear algebra package is to be used for the solution of the augmented linear system (for obtaining the search directions). Note, that MA27, MA57, MA86, and MA97 are included with a commercially supported GAMS/IpoptH license only. To use MA27, MA57, MA86, or MA97 with GAMS/Ipopt, or to use HSL_MA77, a HSL library needs to be provided by the user. To use Pardiso from pardiso-project.org, a Pardiso library needs to be provided by the user. ATTENTION: Before Ipopt 3.14 (GAMS 36), value pardiso specified to use Pardiso from Intel MKL. With GAMS 36, this value has been renamed to pardisomkl. On GAMS systems for ARM64 CPUs, option value pardisomkl is not available.

value	meaning
ma27	IpoptH: use the Harwell routine MA27; Ipopt: load the Harwell routine MA27 from user-provided library
ma57	IpoptH: use the Harwell routine MA57; Ipopt: load the Harwell routine MA57 from user-provided library
ma77	load the Harwell routine HSL_MA77 from user-provided library
ma86	IpoptH: use the Harwell routine HSL_MA86; Ipopt: load the Harwell routine HSL_MA86 from user-provided library
ma97	IpoptH: use the Harwell routine HSL_MA97; Ipopt: load the Harwell routine HSL_MA97 from user-provided library
pardiso	load the Pardiso package from pardiso-project.org from user-provided library at runtime
pardisomkl	use the Pardiso package from Intel MKL
mumps	use the Mumps package

Default: ma27, if IpoptH, otherwise mumps

linear_system_scaling: Method for scaling the linear system. ↵

Determines the method used to compute symmetric scaling factors for the augmented system (see also the "linear_scaling_on_demand" option). This scaling is independent of the NLP problem scaling. Note, that MC19 is included with a commercially supported GAMS/IpoptH license only. To use MC19 with GAMS/Ipopt, a HSL library needs to be provided by the user.

value	meaning
none	no scaling will be performed
mc19	IpoptH: use the Harwell routine MC19; Ipopt: load the Harwell routine MC19 from user-provided library
slack-based	use the slack values

Default: mc19, if IpoptH, otherwise none

ma27_ignore_singularity (advanced): Whether to use MA27's ability to solve a linear system even if the matrix is singular. ↵

Setting this option to "yes" means that Ipopt will call MA27 to compute solutions for right hand sides, even if MA27 has detected that the matrix is singular (but is still able to solve the linear system). In some cases this might be better than using Ipopt's heuristic of small perturbation of the lower diagonal of the KKT matrix.

Range: yes, no

Default: no

ma27_la_init_factor: Real workspace memory for MA27. ↵

The initial real workspace memory = la_init_factor * memory required by unfactored system. Ipopt will increase the workspace size by ma27_meminc_factor if required.

Range: [1, ∞]

Default: 5

ma27_liw_init_factor: Integer workspace memory for MA27. ↵

The initial integer workspace memory = liw_init_factor * memory required by unfactored system. Ipopt will increase the workspace size by ma27_meminc_factor if required.

Range: [1, ∞]

Default: 5

ma27_meminc_factor: Increment factor for workspace size for MA27. ↵

If the integer or real workspace is not large enough, Ipopt will increase its size by this factor.

Range: [1, ∞]

Default: 2

ma27_pivtol: Pivot tolerance for the linear solver MA27. ↵

A smaller number pivots for sparsity, a larger number pivots for stability.

Range: (0, 1)

Default: 1e-08

ma27_pivtolmax: Maximum pivot tolerance for the linear solver MA27. ↵

Ipopt may increase pivtol as high as ma27_pivtolmax to get a more accurate solution to the linear system.

Range: (0, 1)

Default: 0.0001

ma27_print_level: Debug printing level for the linear solver MA27 ↵

0: no printing; 1: Error messages only; 2: Error and warning messages; 3: Error and warning messages and terse monitoring; 4: All information.

Range: {0, ..., 4}

Default: 0

ma27_skip_inertia_check (advanced): Whether to always pretend that inertia is correct. ↵

Setting this option to "yes" essentially disables inertia check. This option makes the algorithm non-robust and easily fail, but it might give some insight into the necessity of inertia control.

Range: yes, no

Default: no

ma57_automatic_scaling: Controls whether to enable automatic scaling in MA57 ↵

For higher reliability of the MA57 solver, you may want to set this option to yes. This is ICNTL(15) in MA57.

Range: yes, no

Default: no

ma57_block_size: Controls block size used by Level 3 BLAS in MA57BD ↵

This is ICNTL(11) in MA57.

Range: {1, ..., ∞}

Default: 16

ma57_node_amalgamation: Node amalgamation parameter ↵

This is ICNTL(12) in MA57.

Range: {1, ..., ∞}

Default: 16

ma57_pivot_order: Controls pivot order in MA57 ↵

This is ICNTL(6) in MA57.

Range: {0, ..., 5}

Default: 5

ma57_pivtol: Pivot tolerance for the linear solver MA57. ↵

A smaller number pivots for sparsity, a larger number pivots for stability.

Range: (0, 1)

Default: 1e-08

ma57_pivtolmax: Maximum pivot tolerance for the linear solver MA57. ↵

Ipopt may increase pivtol as high as ma57_pivtolmax to get a more accurate solution to the linear system.

Range: (0, 1)

Default: 0.0001

ma57_pre_alloc: Safety factor for work space memory allocation for the linear solver MA57. ↵

If 1 is chosen, the suggested amount of work space is used. However, choosing a larger number might avoid reallocation if the suggest values do not suffice.

Range: [1, ∞]

Default: 1.05

ma57_print_level: Debug printing level for the linear solver MA57 ↵

0: no printing; 1: Error messages only; 2: Error and warning messages; 3: Error and warning messages and terse monitoring; ≥4: All information.

Range: {0, ..., ∞}

Default: 0

ma57_small_pivot_flag: Handling of small pivots ↵

If set to 1, then when small entries defined by CNTL(2) are detected they are removed and the corresponding pivots placed at the end of the factorization. This can be particularly efficient if the matrix is highly rank deficient. This is ICNTL(16) in MA57.

Range: {0, ..., 1}

Default: 0

ma77_buffer_lpage: Number of scalars per MA77 in-core buffer page in the out-of-core solver MA77 ↵

Must be at most ma77_file_size.

Range: {1, ..., ∞}

Default: 4096

ma77_buffer_npage: Number of pages that make up MA77 buffer ↵

Number of pages of size buffer_lpage that exist in-core for the out-of-core solver MA77.

Range: {1, ..., ∞}

Default: 1600

ma77_file_size: Target size of each temporary file for MA77, scalars per type ↵

MA77 uses many temporary files, this option controls the size of each one. It is measured in the number of entries (int or double), NOT bytes.

Range: {1, ..., ∞}

Default: 2097152

ma77_maxstore: Maximum storage size for MA77 in-core mode ↵

If greater than zero, the maximum size of factors stored in core before out-of-core mode is invoked.

Range: {0, ..., ∞}

Default: 0

ma77_nemin: Node Amalgamation parameter ↵

Two nodes in elimination tree are merged if result has fewer than ma77_nemin variables.

Range: {1, ..., ∞}

Default: 8

ma77_order: Controls type of ordering used by MA77 ↵

value	meaning
amd	Use the HSL_MC68 approximate minimum degree algorithm
metis	Use the MeTiS nested dissection algorithm (if available)

Default: metis

ma77_print_level: Debug printing level for the linear solver MA77 ↵

<0: no printing; 0: Error and warning messages only; 1: Limited diagnostic printing; >1 Additional diagnostic printing.

Range: {-∞, ..., ∞}

Default: -1

ma77_small: Zero Pivot Threshold ↵

Any pivot less than ma77_small is treated as zero.

Range: [0, ∞]

Default: 1e-20

ma77_static: Static Pivoting Threshold ↵

See MA77 documentation. Either ma77_static=0.0 or ma77_static>ma77_small. ma77_static=0.0 disables static pivoting.

Range: [0, ∞]

Default: 0

ma77_u: Pivoting Threshold ↵

See MA77 documentation.

Range: [0, 0.5]

Default: 1e-08

ma77_umax: Maximum Pivoting Threshold ↵

Maximum value to which u will be increased to improve quality.

Range: [0, 0.5]

Default: 0.0001

ma86_nemin: Node Amalgamation parameter ↵

Two nodes in elimination tree are merged if result has fewer than ma86_nemin variables.

Range: {1, ..., ∞}

Default: 32

ma86_order: Controls type of ordering ↵

value	meaning
auto	Try both AMD and MeTiS, pick best
amd	Use the HSL_MC68 approximate minimum degree algorithm
metis	Use the MeTiS nested dissection algorithm (if available)

Default: auto

ma86_print_level: Debug printing level ↵

<0: no printing; 0: Error and warning messages only; 1: Limited diagnostic printing; >1 Additional diagnostic printing.

Range: {-∞, ..., ∞}

Default: -1

ma86_scaling: Controls scaling of matrix ↵

value	meaning
none	Do not scale the linear system matrix
mc64	Scale linear system matrix using MC64
mc77	Scale linear system matrix using MC77 [1,3,0]

Default: mc64

ma86_small: Zero Pivot Threshold ↵

Any pivot less than ma86_small is treated as zero.

Range: [0, ∞]

Default: 1e-20

ma86_static: Static Pivoting Threshold ↵

See MA86 documentation. Either ma86_static=0.0 or ma86_static>ma86_small. ma86_static=0.0 disables static pivoting.

Range: [0, ∞]

Default: 0

ma86_u: Pivoting Threshold ↵

See MA86 documentation.

Range: [0, 0.5]

Default: 1e-08

ma86_umax: Maximum Pivoting Threshold ↵

Maximum value to which u will be increased to improve quality.

Range: [0, 0.5]

Default: 0.0001

ma97_nemin: Node Amalgamation parameter ↵

Two nodes in elimination tree are merged if result has fewer than ma97_nemin variables.

Range: {1, ..., ∞}

Default: 8

ma97_order: Controls type of ordering ↵

value	meaning
auto	Use HSL_MA97 heuristic to guess best of AMD and METIS
best	Try both AMD and MeTiS, pick best
amd	Use the HSL_MC68 approximate minimum degree algorithm
metis	Use the MeTiS nested dissection algorithm
matched-auto	Use the HSL_MC80 matching with heuristic choice of AMD or METIS
matched-metis	Use the HSL_MC80 matching based ordering with METIS
matched-amd	Use the HSL_MC80 matching based ordering with AMD

Default: auto

ma97_print_level: Debug printing level ↵

<0: no printing; 0: Error and warning messages only; 1: Limited diagnostic printing; >1 Additional diagnostic printing.

Range: {-∞, ..., ∞}

Default: -1

ma97_scaling: Specifies strategy for scaling ↵

value	meaning
none	Do not scale the linear system matrix
mc30	Scale all linear system matrices using MC30
mc64	Scale all linear system matrices using MC64
mc77	Scale all linear system matrices using MC77 [1,3,0]
dynamic	Dynamically select scaling according to rules specified by ma97_scalingX and ma97_switchX options.

Default: dynamic

ma97_scaling1 (advanced): First scaling. ↵

If ma97_scaling=dynamic, this scaling is used according to the trigger ma97_switch1. If ma97_switch2 is triggered it is disabled.

value	meaning
none	No scaling
mc30	Scale linear system matrix using MC30
mc64	Scale linear system matrix using MC64
mc77	Scale linear system matrix using MC77 [1,3,0]

Default: mc64

ma97_scaling2 (advanced): Second scaling. ↵

If ma97_scaling=dynamic, this scaling is used according to the trigger ma97_switch2. If ma97_switch3 is triggered it is disabled.

value	meaning
none	No scaling
mc30	Scale linear system matrix using MC30
mc64	Scale linear system matrix using MC64
mc77	Scale linear system matrix using MC77 [1,3,0]

Default: mc64

ma97_scaling3 (advanced): Third scaling. ↵

If ma97_scaling=dynamic, this scaling is used according to the trigger ma97_switch3.

value	meaning
none	No scaling
mc30	Scale linear system matrix using MC30
mc64	Scale linear system matrix using MC64
mc77	Scale linear system matrix using MC77 [1,3,0]

Default: mc64

ma97_small: Zero Pivot Threshold ↵

Any pivot less than ma97_small is treated as zero.

Range: [0, ∞]

Default: 1e-20

ma97_solve_blas3 (advanced): Controls if blas2 or blas3 routines are used for solve ↵

value	meaning
no	Use BLAS2 (faster, some implementations bit incompatible)
yes	Use BLAS3 (slower)

Default: no

ma97_switch1 (advanced): First switch, determine when ma97_scaling1 is enabled. ↵

If ma97_scaling=dynamic, ma97_scaling1 is enabled according to this condition. If ma97_switch2 occurs this option is henceforth ignored.

value	meaning
never	Scaling is never enabled.
at_start	Scaling to be used from the very start.
at_start_reuse	Scaling to be used on first iteration, then reused thereafter.
on_demand	Scaling to be used after Ipopt request improved solution (i.e. iterative refinement has failed).
on_demand_reuse	As on_demand, but reuse scaling from previous itr
high_delay	Scaling to be used after more than 0.05*n delays are present
high_delay_reuse	Scaling to be used only when previous itr created more that 0.05*n additional delays, otherwise reuse scaling from previous itr
od_hd	Combination of on_demand and high_delay
od_hd_reuse	Combination of on_demand_reuse and high_delay_reuse

Default: od_hd_reuse

ma97_switch2 (advanced): Second switch, determine when ma97_scaling2 is enabled. ↵

If ma97_scaling=dynamic, ma97_scaling2 is enabled according to this condition. If ma97_switch3 occurs this option is henceforth ignored.

value	meaning
never	Scaling is never enabled.
at_start	Scaling to be used from the very start.
at_start_reuse	Scaling to be used on first iteration, then reused thereafter.
on_demand	Scaling to be used after Ipopt request improved solution (i.e. iterative refinement has failed).
on_demand_reuse	As on_demand, but reuse scaling from previous itr
high_delay	Scaling to be used after more than 0.05*n delays are present
high_delay_reuse	Scaling to be used only when previous itr created more that 0.05*n additional delays, otherwise reuse scaling from previous itr
od_hd	Combination of on_demand and high_delay
od_hd_reuse	Combination of on_demand_reuse and high_delay_reuse

Default: never

ma97_switch3 (advanced): Third switch, determine when ma97_scaling3 is enabled. ↵

If ma97_scaling=dynamic, ma97_scaling3 is enabled according to this condition.

value	meaning
never	Scaling is never enabled.
at_start	Scaling to be used from the very start.
at_start_reuse	Scaling to be used on first iteration, then reused thereafter.
on_demand	Scaling to be used after Ipopt request improved solution (i.e. iterative refinement has failed).
on_demand_reuse	As on_demand, but reuse scaling from previous itr
high_delay	Scaling to be used after more than 0.05*n delays are present
high_delay_reuse	Scaling to be used only when previous itr created more that 0.05*n additional delays, otherwise reuse scaling from previous itr
od_hd	Combination of on_demand and high_delay
od_hd_reuse	Combination of on_demand_reuse and high_delay_reuse

Default: never

ma97_u: Pivoting Threshold ↵

See MA97 documentation.

Range: [0, 0.5]

Default: 1e-08

ma97_umax: Maximum Pivoting Threshold ↵

See MA97 documentation.

Range: [0, 0.5]

Default: 0.0001

max_cpu_time: Maximum number of CPU seconds. ↵

A limit on CPU seconds that Ipopt can use to solve one problem. If during the convergence check this limit is exceeded, Ipopt will terminate with a corresponding message.

Range: (0, ∞]

Default: 1e+20

max_filter_resets (advanced): Maximal allowed number of filter resets ↵

A positive number enables a heuristic that resets the filter, whenever in more than "filter_reset_trigger" successive iterations the last rejected trial steps size was rejected because of the filter. This option determine the maximal number of resets that are allowed to take place.

Range: {0, ..., ∞}

Default: 5

max_hessian_perturbation: Maximum value of regularization parameter for handling negative curvature. ↵

In order to guarantee that the search directions are indeed proper descent directions, Ipopt requires that the inertia of the (augmented) linear system for the step computation has the correct number of negative and positive eigenvalues. The idea is that this guides the algorithm away from maximizers and makes Ipopt more likely converge to first order optimal points that are minimizers. If the inertia is not correct, a multiple of the identity matrix is added to the Hessian of the Lagrangian in the augmented system. This parameter gives the maximum value of the regularization parameter. If a regularization of that size is not enough, the algorithm skips this iteration and goes to the restoration phase. This is delta_w^max in the implementation paper.

Range: (0, ∞]

Default: 1e+20

max_iter: Maximum number of iterations. ↵

The algorithm terminates with a message if the number of iterations exceeded this number.

Range: {0, ..., ∞}

Default: GAMS iterlim

max_refinement_steps: Maximum number of iterative refinement steps per linear system solve. ↵

Iterative refinement (on the full unsymmetric system) is performed for each right hand side. This option determines the maximum number of iterative refinement steps.

Range: {0, ..., ∞}

Default: 10

max_resto_iter (advanced): Maximum number of successive iterations in restoration phase. ↵

The algorithm terminates with an error message if the number of iterations successively taken in the restoration phase exceeds this number.

Range: {0, ..., ∞}

Default: 3000000

max_soc: Maximum number of second order correction trial steps at each iteration. ↵

Choosing 0 disables the second order corrections. This is p^{max} of Step A-5.9 of Algorithm A in the implementation paper.

Range: {0, ..., ∞}

Default: 4

max_soft_resto_iters (advanced): Maximum number of iterations performed successively in soft restoration phase. ↵

If the soft restoration phase is performed for more than so many iterations in a row, the regular restoration phase is called.

Range: {0, ..., ∞}

Default: 10

max_wall_time: Maximum number of walltime clock seconds. ↵

A limit on walltime clock seconds that Ipopt can use to solve one problem. If during the convergence check this limit is exceeded, Ipopt will terminate with a corresponding message.

Range: (0, ∞]

Default: GAMS reslim

mehrotra_algorithm: Indicates whether to do Mehrotra's predictor-corrector algorithm. ↵

If enabled, line search is disabled and the (unglobalized) adaptive mu strategy is chosen with the "probing" oracle, and "corrector_type=affine" is used without any safeguards; you should not set any of those options explicitly in addition. Also, unless otherwise specified, the values of "bound_push", "bound_frac", and "bound_mult_init_val" are set more aggressive, and sets "alpha_for_y=bound_mult". The Mehrotra's predictor-corrector algorithm works usually very well for LPs and convex QPs.

Range: yes, no

Default: no

min_hessian_perturbation: Smallest perturbation of the Hessian block. ↵

The size of the perturbation of the Hessian block is never selected smaller than this value, unless no perturbation is necessary. This is delta_w^min in implementation paper.

Range: [0, ∞]

Default: 1e-20

min_refinement_steps: Minimum number of iterative refinement steps per linear system solve. ↵

Iterative refinement (on the full unsymmetric system) is performed for each right hand side. This option determines the minimum number of iterative refinements (i.e. at least "min_refinement_steps" iterative refinement steps are enforced per right hand side.)

Range: {0, ..., ∞}

Default: 1

mu_allow_fast_monotone_decrease (advanced): Allow skipping of barrier problem if barrier test is already met. ↵

value	meaning
no	Take at least one iteration per barrier problem even if the barrier test is already met for the updated barrier parameter
yes	Allow fast decrease of mu if barrier test it met

Default: yes

mu_init: Initial value for the barrier parameter. ↵

This option determines the initial value for the barrier parameter (mu). It is only relevant in the monotone, Fiacco-McCormick version of the algorithm. (i.e., if "mu_strategy" is chosen as "monotone")

Range: (0, ∞]

Default: 0.1

mu_linear_decrease_factor: Determines linear decrease rate of barrier parameter. ↵

For the Fiacco-McCormick update procedure the new barrier parameter mu is obtained by taking the minimum of mu*"mu_linear_decrease_factor" and mu^"superlinear_decrease_power". This is kappa_mu in implementation paper. This option is also used in the adaptive mu strategy during the monotone mode.

Range: (0, 1)

Default: 0.2

mu_max: Maximum value for barrier parameter. ↵

This option specifies an upper bound on the barrier parameter in the adaptive mu selection mode. If this option is set, it overwrites the effect of mu_max_fact. (Only used if option "mu_strategy" is chosen as "adaptive".)

Range: (0, ∞]

Default: 100000

mu_max_fact: Factor for initialization of maximum value for barrier parameter. ↵

This option determines the upper bound on the barrier parameter. This upper bound is computed as the average complementarity at the initial point times the value of this option. (Only used if option "mu_strategy" is chosen as "adaptive".)

Range: (0, ∞]

Default: 1000

mu_min: Minimum value for barrier parameter. ↵

This option specifies the lower bound on the barrier parameter in the adaptive mu selection mode. By default, it is set to the minimum of 1e-11 and min("tol","compl_inf_tol")/("barrier_tol_factor"+1), which should be a reasonable value. (Only used if option "mu_strategy" is chosen as "adaptive".)

Range: (0, ∞]

Default: 1e-11

mu_oracle: Oracle for a new barrier parameter in the adaptive strategy. ↵

Determines how a new barrier parameter is computed in each "free-mode" iteration of the adaptive barrier parameter strategy. (Only considered if "adaptive" is selected for option "mu_strategy").

value	meaning
probing	Mehrotra's probing heuristic
loqo	LOQO's centrality rule
quality-function	minimize a quality function

Default: quality-function

mu_strategy: Update strategy for barrier parameter. ↵

Determines which barrier parameter update strategy is to be used.

value	meaning
monotone	use the monotone (Fiacco-McCormick) strategy
adaptive	use the adaptive update strategy

Default: adaptive

mu_superlinear_decrease_power: Determines superlinear decrease rate of barrier parameter. ↵

For the Fiacco-McCormick update procedure the new barrier parameter mu is obtained by taking the minimum of mu*"mu_linear_decrease_factor" and mu^"superlinear_decrease_power". This is theta_mu in implementation paper. This option is also used in the adaptive mu strategy during the monotone mode.

Range: (1, 2)

Default: 1.5

mu_target: Desired value of complementarity. ↵

Usually, the barrier parameter is driven to zero and the termination test for complementarity is measured with respect to zero complementarity. However, in some cases it might be desired to have Ipopt solve barrier problem for strictly positive value of the barrier parameter. In this case, the value of "mu_target" specifies the final value of the barrier parameter, and the termination tests are then defined with respect to the barrier problem for this value of the barrier parameter.

Range: [0, ∞]

Default: 0

mumps_dep_tol (advanced): Threshold to consider a pivot at zero in detection of linearly dependent constraints with MUMPS. ↵

This is CNTL(3) in MUMPS.

Range: real

Default: 0

mumps_mem_percent: Percentage increase in the estimated working space for MUMPS. ↵

When significant extra fill-in is caused by numerical pivoting, larger values of mumps_mem_percent may help use the workspace more efficiently. On the other hand, if memory requirement are too large at the very beginning of the optimization, choosing a much smaller value for this option, such as 5, might reduce memory requirements.

Range: {0, ..., ∞}

Default: 1000

mumps_permuting_scaling: Controls permuting and scaling in MUMPS ↵

This is ICNTL(6) in MUMPS.

Range: {0, ..., 7}

Default: 7

mumps_pivot_order: Controls pivot order in MUMPS ↵

This is ICNTL(7) in MUMPS.

Range: {0, ..., 7}

Default: 7

mumps_pivtol: Pivot tolerance for the linear solver MUMPS. ↵

A smaller number pivots for sparsity, a larger number pivots for stability.

Range: [0, 1]

Default: 1e-06

mumps_pivtolmax: Maximum pivot tolerance for the linear solver MUMPS. ↵

Ipopt may increase pivtol as high as pivtolmax to get a more accurate solution to the linear system.

Range: [0, 1]

Default: 0.1

mumps_print_level: Debug printing level for the linear solver MUMPS ↵

0: no printing; 1: Error messages only; 2: Error, warning, and main statistic messages; 3: Error and warning messages and terse diagnostics; ≥4: All information.

Range: {0, ..., ∞}

Default: 0

mumps_scaling: Controls scaling in MUMPS ↵

This is ICNTL(8) in MUMPS.

Range: {-2, ..., 77}

Default: 77

neg_curv_test_reg: Whether to do the curvature test with the primal regularization (see Zavala and Chiang, 2014). ↵

value	meaning
yes	use primal regularization with the inertia-free curvature test
no	use original IPOPT approach, in which the primal regularization is ignored

Default: yes

neg_curv_test_tol: Tolerance for heuristic to ignore wrong inertia. ↵

If nonzero, incorrect inertia in the augmented system is ignored, and Ipopt tests if the direction is a direction of positive curvature. This tolerance is alpha_n in the paper by Zavala and Chiang (2014) and it determines when the direction is considered to be sufficiently positive. A value in the range of [1e-12, 1e-11] is recommended.

Range: [0, ∞]

Default: 0

nlp_scaling_constr_target_gradient (advanced): Target value for constraint function gradient size. ↵

If a positive number is chosen, the scaling factors for the constraint functions are computed so that the gradient has the max norm of the given size at the starting point. This overrides nlp_scaling_max_gradient for the constraint functions.

Range: [0, ∞]

Default: 0

nlp_scaling_max_gradient: Maximum gradient after NLP scaling. ↵

This is the gradient scaling cut-off. If the maximum gradient is above this value, then gradient based scaling will be performed. Scaling parameters are calculated to scale the maximum gradient back to this value. (This is g_max in Section 3.8 of the implementation paper.) Note: This option is only used if "nlp_scaling_method" is chosen as "gradient-based".

Range: (0, ∞]

Default: 100

nlp_scaling_method: Select the technique used for scaling the NLP. ↵

Selects the technique used for scaling the problem internally before it is solved. For user-scaling, the parameters come from the NLP.

value	meaning
none	no problem scaling will be performed
gradient-based	scale the problem so the maximum gradient at the starting point is nlp_scaling_max_gradient
equilibration-based	scale the problem so that first derivatives are of order 1 at random points (GAMS/Ipopt: requires user-provided library with HSL routine MC19)

Default: gradient-based if GAMS scaleopt is not set, otherwise none

nlp_scaling_min_value: Minimum value of gradient-based scaling values. ↵

This is the lower bound for the scaling factors computed by gradient-based scaling method. If some derivatives of some functions are huge, the scaling factors will otherwise become very small, and the (unscaled) final constraint violation, for example, might then be significant. Note: This option is only used if "nlp_scaling_method" is chosen as "gradient-based".

Range: [0, ∞]

Default: 1e-08

nlp_scaling_obj_target_gradient (advanced): Target value for objective function gradient size. ↵

If a positive number is chosen, the scaling factor for the objective function is computed so that the gradient has the max norm of the given size at the starting point. This overrides nlp_scaling_max_gradient for the objective function.

Range: [0, ∞]

Default: 0

nu_inc (advanced): Increment of the penalty parameter. ↵

Range: (0, ∞]

Default: 0.0001

nu_init (advanced): Initial value of the penalty parameter. ↵

Range: (0, ∞]

Default: 1e-06

obj_max_inc (advanced): Determines the upper bound on the acceptable increase of barrier objective function. ↵

Trial points are rejected if they lead to an increase in the barrier objective function by more than obj_max_inc orders of magnitude.

Range: (1, ∞]

Default: 5

pardiso_iter_coarse_size (advanced): Maximum Size of Coarse Grid Matrix ↵

DPARM(3)

Range: {1, ..., ∞}

Default: 5000

pardiso_iter_dropping_factor (advanced): dropping value for incomplete factor ↵

DPARM(5)

Range: (0, 1)

Default: 0.5

pardiso_iter_dropping_schur (advanced): dropping value for sparsify schur complement factor ↵

DPARM(6)

Range: (0, 1)

Default: 0.1

pardiso_iter_inverse_norm_factor (advanced): ↵

DPARM(8)

Range: (1, ∞]

Default: 5e+06

pardiso_iter_max_levels (advanced): Maximum Size of Grid Levels ↵

DPARM(4)

Range: {1, ..., ∞}

Default: 10

pardiso_iter_max_row_fill (advanced): max fill for each row ↵

DPARM(7)

Range: {1, ..., ∞}

Default: 10000000

pardiso_iter_relative_tol (advanced): Relative Residual Convergence ↵

DPARM(2)

Range: (0, 1)

Default: 1e-06

pardiso_iterative (advanced): Switch for iterative solver in Pardiso library ↵

Range: yes, no

Default: no

pardiso_matching_strategy: Matching strategy to be used by Pardiso ↵

This is IPAR(13) in Pardiso manual.

value	meaning
complete	Match complete (IPAR(13)=1)
complete+2x2	Match complete+2x2 (IPAR(13)=2)
constraints	Match constraints (IPAR(13)=3)

Default: complete+2x2

pardiso_max_droptol_corrections (advanced): Maximal number of decreases of drop tolerance during one solve. ↵

This is relevant only for iterative Pardiso options.

Range: {1, ..., ∞}

Default: 4

pardiso_max_iter (advanced): Maximum number of Krylov-Subspace Iteration ↵

DPARM(1)

Range: {1, ..., ∞}

Default: 500

pardiso_max_iterative_refinement_steps: Limit on number of iterative refinement steps. ↵

The solver does not perform more than the absolute value of this value steps of iterative refinement and stops the process if a satisfactory level of accuracy of the solution in terms of backward error is achieved. If negative, the accumulation of the residue uses extended precision real and complex data types. Perturbed pivots result in iterative refinement. The solver automatically performs two steps of iterative refinements when perturbed pivots are obtained during the numerical factorization and this option is set to 0.

Range: {-∞, ..., ∞}

Default: 0

pardiso_msglvl: Pardiso message level ↵

This is MSGLVL in the Pardiso manual.

Range: {0, ..., ∞}

Default: 0

pardiso_order: Controls the fill-in reduction ordering algorithm for the input matrix. ↵

value	meaning
amd	minimum degree algorithm
one
metis	MeTiS nested dissection algorithm
pmetis	parallel (OpenMP) version of MeTiS nested dissection algorithm
four
five

Default: metis

pardiso_redo_symbolic_fact_only_if_inertia_wrong (advanced): Toggle for handling case when elements were perturbed by Pardiso. ↵

value	meaning
no	Always redo symbolic factorization when elements were perturbed
yes	Only redo symbolic factorization when elements were perturbed if also the inertia was wrong

Default: no

pardiso_repeated_perturbation_means_singular (advanced): Whether to assume that matrix is singular if elements were perturbed after recent symbolic factorization. ↵

Range: yes, no

Default: no

pardiso_skip_inertia_check (advanced): Whether to pretend that inertia is correct. ↵

Setting this option to "yes" essentially disables inertia check. This option makes the algorithm non-robust and easily fail, but it might give some insight into the necessity of inertia control.

Range: yes, no

Default: no

pardisolib: Name of library containing Pardiso routines (from pardiso-project.org) for load at runtime ↵

Range: string

Default: libpardiso.so (Linux), libpardiso.dylib (macOS), libpardiso.dll (Windows)

pardisomkl_matching_strategy: Matching strategy to be used by Pardiso ↵

This is IPAR(13) in Pardiso manual.

value	meaning
complete	Match complete (IPAR(13)=1)
complete+2x2	Match complete+2x2 (IPAR(13)=2)
constraints	Match constraints (IPAR(13)=3)

Default: complete+2x2

pardisomkl_max_iterative_refinement_steps: Limit on number of iterative refinement steps. ↵

The solver does not perform more than the absolute value of this value steps of iterative refinement and stops the process if a satisfactory level of accuracy of the solution in terms of backward error is achieved. If negative, the accumulation of the residue uses extended precision real and complex data types. Perturbed pivots result in iterative refinement. The solver automatically performs two steps of iterative refinements when perturbed pivots are obtained during the numerical factorization and this option is set to 0.

Range: {-∞, ..., ∞}

Default: 1

pardisomkl_msglvl: Pardiso message level ↵

This is MSGLVL in the Pardiso manual.

Range: {0, ..., ∞}

Default: 0

pardisomkl_order: Controls the fill-in reduction ordering algorithm for the input matrix. ↵

value	meaning
amd	minimum degree algorithm
one	undocumented
metis	MeTiS nested dissection algorithm
pmetis	parallel (OpenMP) version of MeTiS nested dissection algorithm

Default: metis

pardisomkl_redo_symbolic_fact_only_if_inertia_wrong (advanced): Toggle for handling case when elements were perturbed by Pardiso. ↵

value	meaning
no	Always redo symbolic factorization when elements were perturbed
yes	Only redo symbolic factorization when elements were perturbed if also the inertia was wrong

Default: no

pardisomkl_repeated_perturbation_means_singular (advanced): Whether to assume that matrix is singular if elements were perturbed after recent symbolic factorization. ↵

Range: yes, no

Default: no

pardisomkl_skip_inertia_check (advanced): Whether to pretend that inertia is correct. ↵

Setting this option to "yes" essentially disables inertia check. This option makes the algorithm non-robust and easily fail, but it might give some insight into the necessity of inertia control.

Range: yes, no

Default: no

perturb_always_cd (advanced): Active permanent perturbation of constraint linearization. ↵

Enabling this option leads to using the delta_c and delta_d perturbation for the computation of every search direction. Usually, it is only used when the iteration matrix is singular.

Range: yes, no

Default: no

perturb_dec_fact: Decrease factor for x-s perturbation. ↵

The factor by which the perturbation is decreased when a trial value is deduced from the size of the most recent successful perturbation. This is kappa_w^- in the implementation paper.

Range: (0, 1)

Default: 0.333333

perturb_inc_fact: Increase factor for x-s perturbation. ↵

The factor by which the perturbation is increased when a trial value was not sufficient - this value is used for the computation of all perturbations except for the first. This is kappa_w^+ in the implementation paper.

Range: (1, ∞]

Default: 8

perturb_inc_fact_first: Increase factor for x-s perturbation for very first perturbation. ↵

The factor by which the perturbation is increased when a trial value was not sufficient - this value is used for the computation of the very first perturbation and allows a different value for the first perturbation than that used for the remaining perturbations. This is bar_kappa_w^+ in the implementation paper.

Range: (1, ∞]

Default: 100

print_advanced_options (advanced): whether to print also advanced options ↵

Range: yes, no

Default: no

print_eval_error: Switch to enable printing information about function evaluation errors into the GAMS listing file. ↵

Range: no, yes

Default: yes

print_frequency_iter: Determines at which iteration frequency the summarizing iteration output line should be printed. ↵

Summarizing iteration output is printed every print_frequency_iter iterations, if at least print_frequency_time seconds have passed since last output.

Range: {1, ..., ∞}

Default: 1

print_frequency_time: Determines at which time frequency the summarizing iteration output line should be printed. ↵

Summarizing iteration output is printed if at least print_frequency_time seconds have passed since last output and the iteration number is a multiple of print_frequency_iter.

Range: [0, ∞]

Default: 0

print_info_string: Enables printing of additional info string at end of iteration output. ↵

This string contains some insider information about the current iteration. For details, look for "Diagnostic Tags" in the Ipopt documentation.

Range: yes, no

Default: no

print_level: Output verbosity level. ↵

Sets the default verbosity level for console output. The larger this value the more detailed is the output.

Range: {0, ..., 12}

Default: 5

print_options_mode: format in which to print options documentation ↵

value	meaning
text	Ordinary text
latex	LaTeX formatted
doxygen	Doxygen (markdown) formatted

Default: text

print_timing_statistics: Switch to print timing statistics. ↵

If selected, the program will print the time spend for selected tasks. This implies timing_statistics=yes.

Range: yes, no

Default: no

quality_function_balancing_term (advanced): The balancing term included in the quality function for centrality. ↵

This determines whether a term is added to the quality function that penalizes situations where the complementarity is much smaller than dual and primal infeasibilities. Only used if option "mu_oracle" is set to "quality-function".

value	meaning
none	no balancing term is added
cubic	Max(0,Max(dual_inf,primal_inf)-compl)^3

Default: none

quality_function_centrality (advanced): The penalty term for centrality that is included in quality function. ↵

This determines whether a term is added to the quality function to penalize deviation from centrality with respect to complementarity. The complementarity measure here is the xi in the Loqo update rule. Only used if option "mu_oracle" is set to "quality-function".

value	meaning
none	no penalty term is added
log	complementarity * the log of the centrality measure
reciprocal	complementarity * the reciprocal of the centrality measure
cubed-reciprocal	complementarity * the reciprocal of the centrality measure cubed

Default: none

quality_function_max_section_steps: Maximum number of search steps during direct search procedure determining the optimal centering parameter. ↵

The golden section search is performed for the quality function based mu oracle. Only used if option "mu_oracle" is set to "quality-function".

Range: {0, ..., ∞}

Default: 8

quality_function_norm_type (advanced): Norm used for components of the quality function. ↵

Only used if option "mu_oracle" is set to "quality-function".

value	meaning
1-norm	use the 1-norm (abs sum)
2-norm-squared	use the 2-norm squared (sum of squares)
max-norm	use the infinity norm (max)
2-norm	use 2-norm

Default: 2-norm-squared

quality_function_section_qf_tol (advanced): Tolerance for the golden section search procedure determining the optimal centering parameter (in the function value space). ↵

The golden section search is performed for the quality function based mu oracle. Only used if option "mu_oracle" is set to "quality-function".

Range: [0, 1)

Default: 0

quality_function_section_sigma_tol (advanced): Tolerance for the section search procedure determining the optimal centering parameter (in sigma space). ↵

The golden section search is performed for the quality function based mu oracle. Only used if option "mu_oracle" is set to "quality-function".

Range: [0, 1)

Default: 0.01

recalc_y: Tells the algorithm to recalculate the equality and inequality multipliers as least square estimates. ↵

This asks the algorithm to recompute the multipliers, whenever the current infeasibility is less than recalc_y_feas_tol. Choosing yes might be helpful in the quasi-Newton option. However, each recalculation requires an extra factorization of the linear system. If a limited memory quasi-Newton option is chosen, this is used by default.

value	meaning
no	use the Newton step to update the multipliers
yes	use least-square multiplier estimates

Default: no

recalc_y_feas_tol: Feasibility threshold for recomputation of multipliers. ↵

If recalc_y is chosen and the current infeasibility is less than this value, then the multipliers are recomputed.

Range: (0, ∞]

Default: 1e-06

replace_bounds (advanced): Whether all variable bounds should be replaced by inequality constraints ↵

This option must be set for the inexact algorithm.

Range: yes, no

Default: no

report_mininfeas_solution: Switch to report intermediate solution with minimal constraint violation to GAMS if the final solution is not feasible. ↵

This option allows to obtain the most feasible solution found by Ipopt during the iteration process, if it stops at a (locally) infeasible solution, due to a limit (time, iterations, ...), or with a failure in the restoration phase.

Range: no, yes

Default: no

required_infeasibility_reduction: Required reduction of infeasibility before leaving restoration phase. ↵

The restoration phase algorithm is performed, until a point is found that is acceptable to the filter and the infeasibility has been reduced by at least the fraction given by this option.

Range: [0, 1)

Default: 0.9

residual_improvement_factor (advanced): Minimal required reduction of residual test ratio in iterative refinement. ↵

If the improvement of the residual test ratio made by one iterative refinement step is not better than this factor, iterative refinement is aborted.

Range: (0, ∞]

Default: 1

residual_ratio_max (advanced): Iterative refinement tolerance ↵

Iterative refinement is performed until the residual test ratio is less than this tolerance (or until "max_refinement_steps" refinement steps are performed).

Range: (0, ∞]

Default: 1e-10

residual_ratio_singular (advanced): Threshold for declaring linear system singular after failed iterative refinement. ↵

If the residual test ratio is larger than this value after failed iterative refinement, the algorithm pretends that the linear system is singular.

Range: (0, ∞]

Default: 1e-05

resto_failure_feasibility_threshold (advanced): Threshold for primal infeasibility to declare failure of restoration phase. ↵

If the restoration phase is terminated because of the "acceptable" termination criteria and the primal infeasibility is smaller than this value, the restoration phase is declared to have failed. The default value is actually 1e2*tol, where tol is the general termination tolerance.

Range: [0, ∞]

Default: 0

resto_penalty_parameter (advanced): Penalty parameter in the restoration phase objective function. ↵

This is the parameter rho in equation (31a) in the Ipopt implementation paper.

Range: (0, ∞]

Default: 1000

resto_proximity_weight (advanced): Weighting factor for the proximity term in restoration phase objective. ↵

This determines how the parameter zeta in equation (29a) in the implementation paper is computed. zeta here is resto_proximity_weight*sqrt(mu), where mu is the current barrier parameter.

Range: [0, ∞]

Default: 1

rho (advanced): Value in penalty parameter update formula. ↵

Range: (0, 1)

Default: 0.1

s_max (advanced): Scaling threshold for the NLP error. ↵

See paragraph after Eqn. (6) in the implementation paper.

Range: (0, ∞]

Default: 100

s_phi (advanced): Exponent for linear barrier function model in the switching rule. ↵

See Eqn. (19) in the implementation paper.

Range: (1, ∞]

Default: 2.3

s_theta (advanced): Exponent for current constraint violation in the switching rule. ↵

See Eqn. (19) in the implementation paper.

Range: (1, ∞]

Default: 1.1

sigma_max (advanced): Maximum value of the centering parameter. ↵

This is the upper bound for the centering parameter chosen by the quality function based barrier parameter update. Only used if option "mu_oracle" is set to "quality-function".

Range: (0, ∞]

Default: 100

sigma_min (advanced): Minimum value of the centering parameter. ↵

This is the lower bound for the centering parameter chosen by the quality function based barrier parameter update. Only used if option "mu_oracle" is set to "quality-function".

Range: [0, ∞]

Default: 1e-06

skip_corr_if_neg_curv (advanced): Whether to skip the corrector step in negative curvature iteration. ↵

The corrector step is not tried if negative curvature has been encountered during the computation of the search direction in the current iteration. This option is only used if "mu_strategy" is "adaptive". Changing this option is experimental.

Range: yes, no

Default: yes

skip_corr_in_monotone_mode (advanced): Whether to skip the corrector step during monotone barrier parameter mode. ↵

The corrector step is not tried if the algorithm is currently in the monotone mode (see also option "barrier_strategy"). This option is only used if "mu_strategy" is "adaptive". Changing this option is experimental.

Range: yes, no

Default: yes

slack_bound_frac: Desired minimum relative distance from the initial slack to bound. ↵

Determines how much the initial slack variables might have to be modified in order to be sufficiently inside the inequality bounds (together with "slack_bound_push"). (This is kappa_2 in Section 3.6 of implementation paper.)

Range: (0, 0.5]

Default: 0.01

slack_bound_push: Desired minimum absolute distance from the initial slack to bound. ↵

Determines how much the initial slack variables might have to be modified in order to be sufficiently inside the inequality bounds (together with "slack_bound_frac"). (This is kappa_1 in Section 3.6 of implementation paper.)

Range: (0, ∞]

Default: 0.01

slack_move (advanced): Correction size for very small slacks. ↵

Due to numerical issues or the lack of an interior, the slack variables might become very small. If a slack becomes very small compared to machine precision, the corresponding bound is moved slightly. This parameter determines how large the move should be. Its default value is mach_eps^{3/4}. See also end of Section 3.5 in implementation paper - but actual implementation might be somewhat different.

Range: [0, ∞]

Default: 1.81899e-12

soc_method: Ways to apply second order correction ↵

This option determines the way to apply second order correction, 0 is the method described in the implementation paper. 1 is the modified way which adds alpha on the rhs of x and s rows.

Range: {0, ..., 1}

Default: 0

soft_resto_pderror_reduction_factor: Required reduction in primal-dual error in the soft restoration phase. ↵

The soft restoration phase attempts to reduce the primal-dual error with regular steps. If the damped primal-dual step (damped only to satisfy the fraction-to-the-boundary rule) is not decreasing the primal-dual error by at least this factor, then the regular restoration phase is called. Choosing "0" here disables the soft restoration phase.

Range: [0, ∞]

Default: 0.9999

start_with_resto: Whether to switch to restoration phase in first iteration. ↵

Setting this option to "yes" forces the algorithm to switch to the feasibility restoration phase in the first iteration. If the initial point is feasible, the algorithm will abort with a failure.

Range: yes, no

Default: no

tau_min (advanced): Lower bound on fraction-to-the-boundary parameter tau. ↵

This is tau_min in the implementation paper. This option is also used in the adaptive mu strategy during the monotone mode.

Range: (0, 1)

Default: 0.99

theta_max_fact (advanced): Determines upper bound for constraint violation in the filter. ↵

The algorithmic parameter theta_max is determined as theta_max_fact times the maximum of 1 and the constraint violation at initial point. Any point with a constraint violation larger than theta_max is unacceptable to the filter (see Eqn. (21) in the implementation paper).

Range: (0, ∞]

Default: 10000

theta_min_fact (advanced): Determines constraint violation threshold in the switching rule. ↵

The algorithmic parameter theta_min is determined as theta_min_fact times the maximum of 1 and the constraint violation at initial point. The switching rule treats an iteration as an h-type iteration whenever the current constraint violation is larger than theta_min (see paragraph before Eqn. (19) in the implementation paper).

Range: (0, ∞]

Default: 0.0001

timing_statistics: Indicates whether to measure time spend in components of Ipopt and NLP evaluation ↵

The overall algorithm time is unaffected by this option.

Range: yes, no

Default: no

tiny_step_tol (advanced): Tolerance for detecting numerically insignificant steps. ↵

If the search direction in the primal variables (x and s) is, in relative terms for each component, less than this value, the algorithm accepts the full step without line search. If this happens repeatedly, the algorithm will terminate with a corresponding exit message. The default value is 10 times machine precision.

Range: [0, ∞]

Default: 2.22045e-15

tiny_step_y_tol (advanced): Tolerance for quitting because of numerically insignificant steps. ↵

If the search direction in the primal variables (x and s) is, in relative terms for each component, repeatedly less than tiny_step_tol, and the step in the y variables is smaller than this threshold, the algorithm will terminate.

Range: [0, ∞]

Default: 0.01

tol: Desired convergence tolerance (relative). ↵

Determines the convergence tolerance for the algorithm. The algorithm terminates successfully, if the (scaled) NLP error becomes smaller than this value, and if the (absolute) criteria according to "dual_inf_tol", "constr_viol_tol", and "compl_inf_tol" are met. This is epsilon_tol in Eqn. (6) in implementation paper. See also "acceptable_tol" as a second termination criterion. Note, some other algorithmic features also use this quantity to determine thresholds etc.

Range: (0, ∞]

Default: 1e-08

warm_start_bound_frac: same as bound_frac for the regular initializer ↵

Range: (0, 0.5]

Default: 0.001

warm_start_bound_push: same as bound_push for the regular initializer ↵

Range: (0, ∞]

Default: 0.001

warm_start_init_point: Warm-start for initial point ↵

Indicates whether this optimization should use a warm start initialization, where values of primal and dual variables are given (e.g., from a previous optimization of a related problem.)

value	meaning
no	do not use the warm start initialization
yes	use the warm start initialization

Default: yes, if run on modified model instance (e.g., from GUSS), otherwise no

warm_start_mult_bound_push: same as mult_bound_push for the regular initializer ↵

Range: (0, ∞]

Default: 0.001

warm_start_mult_init_max: Maximum initial value for the equality multipliers. ↵

Range: real

Default: 1e+06

warm_start_slack_bound_frac: same as slack_bound_frac for the regular initializer ↵

Range: (0, 0.5]

Default: 0.001

warm_start_slack_bound_push: same as slack_bound_push for the regular initializer ↵

Range: (0, ∞]

Default: 0.001

warm_start_target_mu (advanced): ↵

Experimental!

Range: real

Default: 0

watchdog_shortened_iter_trigger: Number of shortened iterations that trigger the watchdog. ↵

If the number of successive iterations in which the backtracking line search did not accept the first trial point exceeds this number, the watchdog procedure is activated. Choosing "0" here disables the watchdog procedure.

Range: {0, ..., ∞}

Default: 10

watchdog_trial_iter_max: Maximum number of watchdog iterations. ↵

This option determines the number of trial iterations allowed before the watchdog procedure is aborted and the algorithm returns to the stored point.

Range: {1, ..., ∞}

Default: 3