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IPOPT and IPOPTH

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 [6, 9, 10, 11, 12] and the IPOPT web site. Most of the IPOPT documentation in the section was taken from the IPOPT manual [5] .

The linear solver in IPOPT

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 [1, 2] (currently the default), and MKL PARDISO [7, 8] (only Linux, Mac OS X, and Windows). 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 [4], 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 solver or the linear algebra routines (Blas and Lapack). The following table summarizes which options are available on which platform.

Linear Algebra MUMPS MKL PARDISO MA27 MA57 HSL MA86 HSL MA97
Linux parallel serial parallel serial serial parallel parallel
Mac OS X parallel serial parallel serial serial parallel parallel
Solaris serial serial not available serial serial parallel parallel
Windows parallel serial parallel serial serial parallel parallel

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 pardiso_order and pardiso_max_iterative_refinement_steps.

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 }

Using Harwell Subroutine Library routines with GAMS/IPOPT.

GAMS/IPOPT can use the HSL routines MA27, MA28, MA57, HSL_MA77, HSL_MA86, HSL_MA97, MC19, and HSL_MC68 when provided as shared library. By telling IPOPT to use one of these routines (see options linear_solver, linear_system_scaling, nlp_scaling_method, dependency_detector), GAMS/IPOPT attempts to load the required routines from the library libhsl.so (Unix-Systems), libhsl.dylib (MacOS X), or libhsl.dll (Windows), respectively.

The HSL routines are available at http://www.hsl.rl.ac.uk/ipopt. Note that it is your responsibility to ensure that you are entitled to download and use these routines!

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.

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

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 [11]), 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 [11] .

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 [11]). A dash (-) indicates that no regularization was done.

alpha_du

The stepsize for the dual variables ( \(\alpha^z_k\) in Eq. (14c) of [11]).

alpha_pr

The stepsize for the primal variables ( \(\alpha_k\) in Eq. (14a) of [11]). 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 [11]) 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 feasible point.

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 the 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 CPU time exceeded.

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

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 or does not satisfy the constraint qualification, or if an external function in GAMS provides incorrect derivative information.

Error in step computation (regularization becomes too large?)!

This messages is printed if IPOPT is unable to compute a search direction, despite several attempts to modify the iteration matrix. Usually, the value of the regularization parameter then becomes too large.

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).

Not enough memory.

An error occurred while trying to allocate memory. The problem may be too large for your current memory and swap configuration.

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 suppo.nosp@m.rt@g.nosp@m.ams.c.nosp@m.om).

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 [11]
  • A: Current iteration is acceptable (alternate termination)
  • a: Perturbation for PD Singularity can't be done, assume singular, see Sec. 3.1 in [11]
  • C: Second Order Correction taken, see Sec. 2.4 in [11]
  • Dh: Hessian degenerate based on multiple iterations, see Sec. 3.1 in [11]
  • Dhj: Hessian/Jacobian degenerate based on multiple iterations, see Sec. 3.1 in [11]
  • Dj: Jacobian degenerate based on multiple iterations, see Sec. 3.1 in [11]
  • dx: \(\delta_x\) perturbation too large, see Sec. 3.1 in [11]
  • 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 [11]
  • F+: Resetting filter due to last few rejections of filter, see Sec. 2.3 in [11]
  • L: Degenerate Jacobian, \(\delta_c\) already perturbed, see Sec. 3.1 in [11]
  • l: Degenerate Jacobian, \(\delta_c\) perturbed, see Sec. 3.1 in [11]
  • M: Magic step taken for slack variables (in backtracking line search)
  • Nh: Hessian not yet degenerate, see Sec. 3.1 in [11]
  • Nhj: Hessian/Jacobian not yet degenerate, see Sec. 3.1 in [11]
  • Nj: Jacobian not yet degenerate, see Sec. 3.1 in [11]
  • NW: Warm start initialization failed (in Warm Start Initialization)
  • q: PD system possibly singular, attempt to improve solution quality, see Sec. 3.1 in [11]
  • R: Solution of restoration phase, see Sec. 3.3 in [11]
  • S: PD system possibly singular, accept current solution, see Sec. 3.1 in [11]
  • s: PD system singular, see Sec. 3.1 in [11]
  • s: Square Problem. Set multipliers to zero (default initialization routine)
  • Tmax: Trial θ is larger than θmax (filter parameter, Eq. (21) in [11])
  • W: Watchdog line search procedure successful, see Sec. 3.2 in [11]
  • w: Watchdog line search procedure unsuccessful, stopped, see Sec. 3.2 in [11]
  • Wb: Undoing most recent SR1 update, see Sec. 5.4.1 in [3]
  • We: Skip Limited-Memory Update in restoration phase, see Sec. 5.4.1 in [3]
  • Wp: Safeguard \(B^0 = \sigma I\) for Limited-Memory Update, see Sec. 5.4.1 in [3]
  • Wr: Resetting Limited-Memory Update, see Sec. 5.4.1 in [3]
  • Ws: Skip Limited-Memory Update since \(s^Ty\) is not positive, see Sec. 5.4.1 in [3]
  • WS: Skip Limited-Memory Update since \(\Delta x\) is too small, see Sec. 5.4.1 in [3]
  • 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

Barrier Parameter Update

Option Description Default
adaptive_mu_globalization Globalization strategy for the adaptive mu selection mode. obj-constr-filter
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_kkt_norm_type Norm used for the KKT error in the adaptive mu globalization strategies. 2-norm-squared
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 iterate should be restored if the monotone mode is entered. no
barrier_tol_factor Factor for mu in barrier stop test. 10
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
fixed_mu_oracle Oracle for the barrier parameter when switching to fixed mode. average_compl
mu_allow_fast_monotone_decrease Allow skipping of barrier problem if barrier test is already met. yes
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_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_max_section_steps Maximum number of search steps during direct search procedure determining the optimal centering parameter. 8
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

Convergence

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. 15
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 violation. 0.0001
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. 1000
max_iter Maximum number of iterations. maxint
mu_target Desired value of complementarity. 0
s_max Scaling threshold for the NLP error. 100
tol Desired convergence tolerance (relative). 1e-08

Hessian Approximation

Option Description Default
hessian_approximation Indicates what Hessian information is to be used. exact
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
limited_memory_initialization Initialization strategy for the limited memory quasi-Newton approximation. scalar1
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_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 approximation. bfgs

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

Line Search

Option Description Default
accept_after_max_steps Accept a trial point after maximal this number of steps. -1
accept_every_trial_step Always accept the first trial step. no
alpha_for_y Method to determine the step size for constraint multipliers. primal
alpha_for_y_tol Tolerance for switching to full equality multiplier steps. 10
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 (unsupported!). 1
corrector_type The type of corrector steps that should be taken (unsupported!). 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
max_soc Maximum number of second order correction trial steps at each iteration. 4
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
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
rho Value in penalty parameter update formula. 0.1
skip_corr_if_neg_curv Skip the corrector step in negative curvature iteration (unsupported!). yes
skip_corr_in_monotone_mode Skip the corrector step during monotone barrier parameter mode (unsupported!). yes
slack_move Correction size for very small slacks. 1.81899e-12
soc_method Ways to apply second order correction 0
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
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
watchdog_shortened_iter_trigger Number of shortened iterations that trigger the watchdog. 10
watchdog_trial_iter_max Maximum number of watchdog iterations. 3

Linear Solver

Option Description Default
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
linear_system_scaling Method for scaling the linear system. mc19

MA27 Linear Solver

Option Description Default
ma27_ignore_singularity Enables MA27's ability to solve a linear system even if the matrix is singular. no
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_skip_inertia_check Always pretend inertia is correct. no

MA28 Linear Solver

Option Description Default
ma28_pivtol Pivot tolerance for linear solver MA28. 0.01

MA57 Linear Solver

Option Description Default
ma57_automatic_scaling Controls MA57 automatic scaling 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_small_pivot_flag 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. 0

MA77 Linear Solver

Option Description Default
ma77_buffer_lpage Number of scalars per MA77 buffer page 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 HSL_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 used by HSL_MA86 auto
ma86_print_level Debug printing level for the linear solver MA86 -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 used by HSL_MA97 auto
ma97_print_level Debug printing level for the linear solver MA97 0
ma97_scaling Specifies strategy for scaling in HSL_MA97 linear solver dynamic
ma97_scaling1 First scaling. mc64
ma97_scaling2 Second scaling. mc64
ma97_scaling3 Third scaling. mc64
ma97_small Zero Pivot Threshold 1e-20
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
ma97_u Pivoting Threshold 1e-08
ma97_umax Maximum Pivoting Threshold 0.0001

Mumps Linear Solver

Option Description Default
mumps_dep_tol Pivot threshold for detection of linearly dependent constraints in MUMPS. 0
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_scaling Controls scaling in MUMPS 77

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
dependency_detection_with_rhs Indicates if the right hand sides of the constraints should be considered during dependency detection no
dependency_detector Indicates which linear solver should be used to detect linearly dependent equality constraints. none
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. yes
jac_c_constant Indicates whether all equality constraints are linear no
jac_d_constant Indicates whether all inequality constraints are linear no
kappa_d Weight for linear damping term (to handle one-sided bounds). 1e-05

NLP Scaling

Option Description Default
nlp_scaling_constr_target_gradient Target value for constraint function gradient size. 0
nlp_scaling_max_gradient Maximum gradient after NLP scaling. 100
nlp_scaling_method Select the technique used for scaling the NLP. gradient-based
nlp_scaling_min_value Minimum value of gradient-based scaling values. 1e-08
nlp_scaling_obj_target_gradient Target value for objective function gradient size. 0

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_timing_statistics Switch to print timing statistics. no
replace_bounds Indicates if all variable bounds should be replaced by inequality constraints no
report_mininfeas_solution Switch to report intermediate solution with minimal constraint violation to GAMS if the final solution is not feasible. no

Pardiso 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. 1
pardiso_msglvl Pardiso message level 0
pardiso_order Controls the fill-in reduction ordering algorithm for the input matrix. metis
pardiso_redo_symbolic_fact_only_if_inertia_wrong Toggle for handling case when elements were perturbed by Pardiso. no
pardiso_repeated_perturbation_means_singular Interpretation of perturbed elements. no
pardiso_skip_inertia_check Always pretend inertia is correct. no

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
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
required_infeasibility_reduction Required reduction of infeasibility before leaving restoration phase. 0.9
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
soft_resto_pderror_reduction_factor Required reduction in primal-dual error in the soft restoration phase. 0.9999
start_with_resto Tells algorithm to switch to restoration phase in first iteration. no

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_exponent Exponent for mu in the regularization for rank-deficient constraint Jacobians. 0.25
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 if we want to do Mehrotra's 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_always_cd Active permanent perturbation of constraint linearization. no
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
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

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 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

Detailed Options Description

acceptable_compl_inf_tol (real): '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.

Default: 0.01

acceptable_constr_viol_tol (real): '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.

Default: 0.01

acceptable_dual_inf_tol (real): '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.

Default: 1e+10

acceptable_iter (integer): 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.

Default: 15

acceptable_obj_change_tol (real): '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.

Default: 1e+20

acceptable_tol (real): '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.

Default: 1e-06

accept_after_max_steps (integer): Accept a trial point after maximal this number of steps.

Even if it does not satisfy line search conditions.

Range: [-1, ∞]

Default: -1

accept_every_trial_step (string): 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.

Default: no

valuemeaning
no don't arbitrarily accept the full step
yes always accept the full step

adaptive_mu_globalization (string): 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".)

Default: obj-constr-filter

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

adaptive_mu_kkterror_red_fact (real): 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 (integer): 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.

Default: 4

adaptive_mu_kkt_norm_type (string): 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 options is also used in the QualityFunctionMuOracle.

Default: 2-norm-squared

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

adaptive_mu_monotone_init_factor (real): 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".

Default: 0.8

adaptive_mu_restore_previous_iterate (string): Indicates if the previous 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).

Default: no

valuemeaning
no don't restore accepted iterate
yes restore accepted iterate

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

This option determines how the step size (alpha_y) will be calculated when updating the constraint multipliers.

Default: primal

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

alpha_for_y_tol (real): 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.

Default: 10

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

(This is gamma_alpha in Eqn. (20) in the implementation paper.)

Range: [0, 1]

Default: 0.05

alpha_red_factor (real): 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 (real): 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).

Default: 10

bound_frac (real): 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 (string): 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, the 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.

Default: constant

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

bound_mult_init_val (real): Initial value for the bound multipliers.

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

Default: 1

bound_mult_reset_threshold (real): 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.

Default: 1000

bound_push (real): 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.)

Default: 0.01

bound_relax_factor (real): 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. If it is set to zero, then then bounds relaxation is disabled. (See Eqn.(35) in implementation paper.)

Default: 1e-10

check_derivatives_for_naninf (string): 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_MORE_DETAILED; so beware of large output!

Default: no

valuemeaning
no Don't check (faster).
yes Check Jacobians and Hessian for Nan and Inf.

compl_inf_tol (real): 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.

Default: 0.0001

constraint_violation_norm_type (string): 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.

Default: 1-norm

valuemeaning
1-norm use the 1-norm
2-norm use the 2-norm
max-norm use the infinity norm

constr_mult_init_max (real): 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.

Default: 1000

constr_mult_reset_threshold (real): 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.

Default: 0

constr_viol_tol (real): Desired threshold for the constraint violation.

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

Default: 0.0001

corrector_compl_avrg_red_fact (real): Complementarity tolerance factor for accepting corrector step (unsupported!).

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

Default: 1

corrector_type (string): The type of corrector steps that should be taken (unsupported!).

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

Default: none

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

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

(See Eqn. (19) in the implementation paper.)

Default: 1

dependency_detection_with_rhs (string): Indicates if the right hand sides of the constraints should be considered during dependency detection

Default: no

valuemeaning
no only look at gradients
yes also consider right hand side

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

The default and available choices depend on how Ipopt has been compiled. This is experimental and does not work well.

Default: none

valuemeaning
ma28 use MA28
mumps use MUMPS
none don't check; no extra work at beginning

diverging_iterates_tol (real): 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.

Default: 1e+20

dual_inf_tol (real): 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.

Default: 1

eta_phi (real): 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 (string): Determines if the original objective function should be evaluated at restoration phase trial points.

Setting this option to "yes" 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.

Default: yes

valuemeaning
no skip evaluation
yes evaluate at every trial point

expect_infeasible_problem (string): 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.

Default: no

valuemeaning
no the problem probably be feasible
yes the problem has a good chance to be infeasible

expect_infeasible_problem_ctol (real): 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.

Default: 0.001

expect_infeasible_problem_ytol (real): 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.

Default: 1e+08

fast_step_computation (string): Indicates if the linear system should be solved quickly.

If set to yes, 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, and the computation of the search direction is a little faster.

Default: no

valuemeaning
no Verify solution of linear system by computing residuals.
yes Trust that linear systems are solved well.

filter_margin_fact (real): 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 (real): Maximum width of margin in obj-constr-filter adaptive globalization strategy.

Default: 1

filter_reset_trigger (integer): 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 (real): 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.)

Default: 0.0001

fixed_mu_oracle (string): 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".)

Default: average_compl

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

fixed_variable_treatment (string): 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" 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 both "make_constraints" and "relax_bounds", bound multipliers are computed for the fixed variables.

Default: make_parameter

valuemeaning
make_constraint Add equality constraints fixing variables
make_parameter Remove fixed variable from optimization variables
relax_bounds Relax fixing bound constraints

gamma_phi (real): 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 (real): 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 (string): 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.

Default: exact

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

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

Default: nonlinear-variables

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

honor_original_bounds (string): 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.

Default: yes

valuemeaning
no Leave final point unchanged
yes Project final point back into original bounds

inf_pr_output (string): 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.

Default: original

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

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

(This is kappa_c in the implementation paper.)

Default: 0.25

jacobian_regularization_value (real): Size of the regularization for rank-deficient constraint Jacobians.

(This is bar delta_c in the implementation paper.)

Default: 1e-08

jac_c_constant (string): Indicates whether all equality constraints are linear

Activating this option will cause Ipopt to ask for the Jacobian of the equality constraints only once from the NLP and reuse this information later.

Default: no

valuemeaning
no Don't assume that all equality constraints are linear
yes Assume that equality constraints Jacobian are constant

jac_d_constant (string): Indicates whether all inequality constraints are linear

Activating this option will cause Ipopt to ask for the Jacobian of the inequality constraints only once from the NLP and reuse this information later.

Default: no

valuemeaning
no Don't assume that all inequality constraints are linear
yes Assume that equality constraints Jacobian are constant

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

(see Section 3.7 in implementation paper.)

Default: 1e-05

kappa_sigma (real): 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.

Default: 1e+10

kappa_soc (real): 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.)

Default: 0.99

least_square_init_duals (string): 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".

Default: no

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

least_square_init_primal (string): 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.

Default: no

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

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

Default: sherman-morrison

valuemeaning
extended use an extended augmented system
sherman-morrison use Sherman-Morrison formula

limited_memory_initialization (string): 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.

Default: scalar1

valuemeaning
constant sigma = limited_memory_init_val
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

limited_memory_init_val (real): 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".

Default: 1

limited_memory_init_val_max (real): 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".

Default: 1e+08

limited_memory_init_val_min (real): 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".

Default: 1e-08

limited_memory_max_history (integer): 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.

Default: 6

limited_memory_max_skipping (integer): Threshold for successive iterations where update is skipped.

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

Range: [1, ∞]

Default: 2

limited_memory_special_for_resto (string): 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".

Default: no

valuemeaning
no use the same update as in regular iterations
yes use the a special update during restoration phase

limited_memory_update_type (string): Quasi-Newton update formula for the limited memory approximation.

Determines which update formula is to be used for the limited-memory quasi-Newton approximation.

Default: bfgs

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

linear_scaling_on_demand (string): 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.

Default: yes

valuemeaning
no Always scale the linear system.
yes Start using linear system scaling if solutions seem not good.

linear_solver (string): 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 only available with a commercially supported GAMS/IpoptH license, or when the user provides a library with HSL code separately. If no GAMS/IpoptH license is available, the default linear solver is MUMPS. Pardiso is only available on Linux and Windows systems. For using Pardiso on non-Linux/Windows systems or MA77, a Pardiso or HSL library need to be provided.

Default: ma27

valuemeaning
ma27 use the Harwell routine MA27
ma57 use the Harwell routine MA57
ma77 use the Harwell routine HSL_MA77
ma86 use the Harwell routine HSL_MA86
ma97 use the Harwell routine HSL_MA97
mumps use MUMPS package
pardiso use the Pardiso package

linear_system_scaling (string): 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. By default, MC19 is only used if MA27 or MA57 are selected as linear solvers. Note, that MC19 is only available with a commercially supported GAMS/IpoptH license, or when the user provides a library with HSL code separately. If no commerical GAMS/IpoptH license is available, the default scaling method is slack-based.

Default: mc19

valuemeaning
mc19 use the Harwell routine MC19
none no scaling will be performed
slack-based use the slack values

line_search_method (string): 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.

Default: filter

valuemeaning
cg-penalty Chen-Goldfarb penalty function
filter Filter method
penalty Standard penalty function

ma27_ignore_singularity (string): Enables 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.

Default: no

valuemeaning
no Don't have MA27 solve singular systems
yes Have MA27 solve singular systems

ma27_la_init_factor (real): 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 meminc_factor if required. This option is only available if Ipopt has been compiled with MA27.

Range: [1, ∞]

Default: 5

ma27_liw_init_factor (real): 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 meminc_factor if required. This option is only available if Ipopt has been compiled with MA27.

Range: [1, ∞]

Default: 5

ma27_meminc_factor (real): 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. This option is only available if Ipopt has been compiled with MA27.

Range: [1, ∞]

Default: 2

ma27_pivtol (real): Pivot tolerance for the linear solver MA27.

A smaller number pivots for sparsity, a larger number pivots for stability. This option is only available if Ipopt has been compiled with MA27.

Range: [0, 1]

Default: 1e-08

ma27_pivtolmax (real): Maximum pivot tolerance for the linear solver MA27.

Ipopt may increase pivtol as high as pivtolmax to get a more accurate solution to the linear system. This option is only available if Ipopt has been compiled with MA27.

Range: [0, 1]

Default: 0.0001

ma27_skip_inertia_check (string): Always pretend 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.

Default: no

valuemeaning
no check inertia
yes skip inertia check

ma28_pivtol (real): Pivot tolerance for linear solver MA28.

This is used when MA28 tries to find the dependent constraints.

Range: [0, 1]

Default: 0.01

ma57_automatic_scaling (string): Controls MA57 automatic scaling

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

Default: no

valuemeaning
no Do not scale the linear system matrix
yes Scale the linear system matrix

ma57_block_size (integer): Controls block size used by Level 3 BLAS in MA57BD

This is ICNTL(11) in MA57.

Range: [1, ∞]

Default: 16

ma57_node_amalgamation (integer): Node amalgamation parameter

This is ICNTL(12) in MA57.

Range: [1, ∞]

Default: 16

ma57_pivot_order (integer): Controls pivot order in MA57

This is ICNTL(6) in MA57.

Range: [0, 5]

Default: 5

ma57_pivtol (real): Pivot tolerance for the linear solver MA57.

A smaller number pivots for sparsity, a larger number pivots for stability. This option is only available if Ipopt has been compiled with MA57.

Range: [0, 1]

Default: 1e-08

ma57_pivtolmax (real): 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. This option is only available if Ipopt has been compiled with MA57.

Range: [0, 1]

Default: 0.0001

ma57_pre_alloc (real): 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. This option is only available if Ipopt has been compiled with MA57.

Range: [1, ∞]

Default: 1.05

ma57_small_pivot_flag (integer): 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 (integer): Number of scalars per MA77 buffer page

Number of scalars per an in-core buffer in the out-of-core solver MA77. Must be at most ma77_file_size.

Range: [1, ∞]

Default: 4096

ma77_buffer_npage (integer): 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 (integer): 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 (integer): 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.

Default: 0

ma77_nemin (integer): Node Amalgamation parameter

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

Range: [1, ∞]

Default: 8

ma77_order (string): Controls type of ordering used by HSL_MA77

This option controls ordering for the solver HSL_MA77.

Default: metis

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

ma77_print_level (integer): Debug printing level for the linear solver MA77

Range: [-∞, ∞]

Default: -1

ma77_small (real): Zero Pivot Threshold

Any pivot less than ma77_small is treated as zero.

Default: 1e-20

ma77_static (real): Static Pivoting Threshold

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

Default: 0

ma77_u (real): Pivoting Threshold

See MA77 documentation.

Range: [0, 0.5]

Default: 1e-08

ma77_umax (real): Maximum Pivoting Threshold

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

Range: [0, 0.5]

Default: 0.0001

ma86_nemin (integer): Node Amalgamation parameter

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

Range: [1, ∞]

Default: 32

ma86_order (string): Controls type of ordering used by HSL_MA86

This option controls ordering for the solver HSL_MA86.

Default: auto

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

ma86_print_level (integer): Debug printing level for the linear solver MA86

Range: [-∞, ∞]

Default: -1

ma86_scaling (string): Controls scaling of matrix

This option controls scaling for the solver HSL_MA86.

Default: mc64

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

ma86_small (real): Zero Pivot Threshold

Any pivot less than ma86_small is treated as zero.

Default: 1e-20

ma86_static (real): Static Pivoting Threshold

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

Default: 0

ma86_u (real): Pivoting Threshold

See MA86 documentation.

Range: [0, 0.5]

Default: 1e-08

ma86_umax (real): Maximum Pivoting Threshold

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

Range: [0, 0.5]

Default: 0.0001

ma97_nemin (integer): Node Amalgamation parameter

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

Range: [1, ∞]

Default: 8

ma97_order (string): Controls type of ordering used by HSL_MA97

Default: auto

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

ma97_print_level (integer): Debug printing level for the linear solver MA97

Range: [-∞, ∞]

Default: 0

ma97_scaling (string): Specifies strategy for scaling in HSL_MA97 linear solver

Default: dynamic

valuemeaning
dynamic Dynamically select scaling according to rules specified by ma97_scalingX and ma97_switchX options.
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]
none Do not scale the linear system matrix

ma97_scaling1 (string): First scaling.

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

Default: mc64

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

ma97_scaling2 (string): Second scaling.

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

Default: mc64

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

ma97_scaling3 (string): Third scaling.

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

Default: mc64

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

ma97_small (real): Zero Pivot Threshold

Any pivot less than ma97_small is treated as zero.

Default: 1e-20

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

Default: no

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

ma97_switch1 (string): 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.

Default: od_hd_reuse

valuemeaning
at_start Scaling to be used from the very start.
at_start_reuse Scaling to be used on first iteration, then reused thereafter.
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
never Scaling is never enabled.
od_hd Combination of on_demand and high_delay
od_hd_reuse Combination of on_demand_reuse and high_delay_reuse
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

ma97_switch2 (string): 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.

Default: never

valuemeaning
at_start Scaling to be used from the very start.
at_start_reuse Scaling to be used on first iteration, then reused thereafter.
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
never Scaling is never enabled.
od_hd Combination of on_demand and high_delay
od_hd_reuse Combination of on_demand_reuse and high_delay_reuse
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

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

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

Default: never

valuemeaning
at_start Scaling to be used from the very start.
at_start_reuse Scaling to be used on first iteration, then reused thereafter.
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
never Scaling is never enabled.
od_hd Combination of on_demand and high_delay
od_hd_reuse Combination of on_demand_reuse and high_delay_reuse
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

ma97_u (real): Pivoting Threshold

See MA97 documentation.

Range: [0, 0.5]

Default: 1e-08

ma97_umax (real): Maximum Pivoting Threshold

See MA97 documentation.

Range: [0, 0.5]

Default: 0.0001

max_cpu_time (real): 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 error message.

Default: 1000

max_filter_resets (integer): 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.

Default: 5

max_hessian_perturbation (real): 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.)

Default: 1e+20

max_iter (integer): Maximum number of iterations.

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

Default: maxint

max_refinement_steps (integer): 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.

Default: 10

max_resto_iter (integer): 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.

Default: 3000000

max_soc (integer): 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.)

Default: 4

max_soft_resto_iters (integer): 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.

Default: 10

mehrotra_algorithm (string): Indicates if we want to do Mehrotra's algorithm.

If set to yes, Ipopt runs as Mehrotra's predictor-corrector algorithm. This works usually very well for LPs and convex QPs. This automatically disables the line search, and chooses the (unglobalized) adaptive mu strategy with the "probing" oracle, and uses "corrector_type=affine" 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".

Default: no

valuemeaning
no Do the usual Ipopt algorithm.
yes Do Mehrotra's predictor-corrector algorithm.

min_hessian_perturbation (real): 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.)

Default: 1e-20

min_refinement_steps (integer): 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.)

Default: 1

mumps_dep_tol (real): Pivot threshold for detection of linearly dependent constraints in MUMPS.

When MUMPS is used to determine linearly dependent constraints, this is determines the threshold for a pivot to be considered zero. This is CNTL(3) in MUMPS.

Range: [-∞, ∞]

Default: 0

mumps_mem_percent (integer): Percentage increase in the estimated working space for MUMPS.

In 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.

Default: 1000

mumps_permuting_scaling (integer): Controls permuting and scaling in MUMPS

This is ICNTL(6) in MUMPS.

Range: [0, 7]

Default: 7

mumps_pivot_order (integer): Controls pivot order in MUMPS

This is ICNTL(7) in MUMPS.

Range: [0, 7]

Default: 7

mumps_pivtol (real): Pivot tolerance for the linear solver MUMPS.

A smaller number pivots for sparsity, a larger number pivots for stability. This option is only available if Ipopt has been compiled with MUMPS.

Range: [0, 1]

Default: 1e-06

mumps_pivtolmax (real): 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. This option is only available if Ipopt has been compiled with MUMPS.

Range: [0, 1]

Default: 0.1

mumps_scaling (integer): Controls scaling in MUMPS

This is ICNTL(8) in MUMPS.

Range: [-2, 77]

Default: 77

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

If set to "no", the algorithm enforces at least one iteration per barrier problem, even if the barrier test is already met for the updated barrier parameter.

Default: yes

valuemeaning
no Take at least one iteration per barrier problem
yes Allow fast decrease of mu if barrier test it met

mu_init (real): 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")

Default: 0.1

mu_linear_decrease_factor (real): 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 (real): 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".)

Default: 100000

mu_max_fact (real): 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".)

Default: 1000

mu_min (real): 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".)

Default: 1e-11

mu_oracle (string): 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").

Default: quality-function

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

mu_strategy (string): Update strategy for barrier parameter.

Determines which barrier parameter update strategy is to be used.

Default: adaptive

valuemeaning
adaptive use the adaptive update strategy
monotone use the monotone (Fiacco-McCormick) strategy

mu_superlinear_decrease_power (real): 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 (real): 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.

Default: 0

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

Default: yes

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

neg_curv_test_tol (real): 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.

Default: 0

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

If a positive number is chosen, the scaling factor the constraint functions 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 constraint functions.

Default: 0

nlp_scaling_max_gradient (real): 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".

Default: 100

nlp_scaling_method (string): 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. If you are using AMPL, they can be specified through suffixes ("scaling_factor")

Default: gradient-based

valuemeaning
equilibration-based scale the problem so that first derivatives are of order 1 at random points (only available with MC19)
gradient-based scale the problem so the maximum gradient at the starting point is scaling_max_gradient
none no problem scaling will be performed

nlp_scaling_min_value (real): 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".

Default: 1e-08

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

If a positive number is chosen, the scaling factor 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.

Default: 0

nu_inc (real): Increment of the penalty parameter.

Default: 0.0001

nu_init (real): Initial value of the penalty parameter.

Default: 1e-06

obj_max_inc (real): 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_matching_strategy (string): Matching strategy to be used by Pardiso

This is IPAR(13) in Pardiso manual.

Default: complete+2x2

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

pardiso_max_iterative_refinement_steps (integer): 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

pardiso_msglvl (integer): Pardiso message level

This determines the amount of analysis output from the Pardiso solver. This is MSGLVL in the Pardiso manual.

Default: 0

pardiso_order (string): Controls the fill-in reduction ordering algorithm for the input matrix.

Default: metis

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

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

Default: no

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

pardiso_repeated_perturbation_means_singular (string): Interpretation of perturbed elements.

Default: no

valuemeaning
no Don't assume that matrix is singular if elements were perturbed after recent symbolic factorization
yes Assume that matrix is singular if elements were perturbed after recent symbolic factorization

pardiso_skip_inertia_check (string): Always pretend 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.

Default: no

valuemeaning
no check inertia
yes skip inertia check

perturb_always_cd (string): Active permanent perturbation of constraint linearization.

This options makes the delta_c and delta_d perturbation be used for the computation of every search direction. Usually, it is only used when the iteration matrix is singular.

Default: no

valuemeaning
no perturbation only used when required
yes always use perturbation

perturb_dec_fact (real): 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 (real): 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 (real): 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 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_eval_error (string): Switch to enable printing information about function evaluation errors into the GAMS listing file.

Default: yes

valuemeaning
no
yes

print_frequency_iter (integer): 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 (real): 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.

Default: 0

print_info_string (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.

Default: no

valuemeaning
no don't print string
yes print string at end of each iteration output

print_level (integer): 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_timing_statistics (string): Switch to print timing statistics.

If selected, the program will print the CPU usage (user time) for selected tasks.

Default: no

valuemeaning
no don't print statistics
yes print all timing statistics

quality_function_balancing_term (string): 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".)

Default: none

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

quality_function_centrality (string): 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".)

Default: none

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

quality_function_max_section_steps (integer): 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".)

Default: 8

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

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

Default: 2-norm-squared

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

quality_function_section_qf_tol (real): 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 (real): 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 (string): 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.

Default: no

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

recalc_y_feas_tol (real): 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.

Default: 1e-06

replace_bounds (string): Indicates if all variable bounds should be replaced by inequality constraints

This option must be set for the inexact algorithm

Default: no

valuemeaning
no leave bounds on variables
yes replace variable bounds by inequality constraints

report_mininfeas_solution (string): 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.

Default: no

valuemeaning
no
yes

required_infeasibility_reduction (real): 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 (real): 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.

Default: 1

residual_ratio_max (real): 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).

Default: 1e-10

residual_ratio_singular (real): 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.

Default: 1e-05

resto_failure_feasibility_threshold (real): 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 1e2*tol, where tol is the general termination tolerance.

Default: 0

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

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

Default: 1000

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

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

Default: 1

rho (real): Value in penalty parameter update formula.

Range: [0, 1]

Default: 0.1

sigma_max (real): 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".)

Default: 100

sigma_min (real): 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".)

Default: 1e-06

skip_corr_if_neg_curv (string): Skip the corrector step in negative curvature iteration (unsupported!).

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".

Default: yes

valuemeaning
no don't skip
yes skip

skip_corr_in_monotone_mode (string): Skip the corrector step during monotone barrier parameter mode (unsupported!).

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".

Default: yes

valuemeaning
no don't skip
yes skip

slack_bound_frac (real): 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 (real): 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.)

Default: 0.01

slack_move (real): 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.)

Default: 1.81899e-12

soc_method (integer): Ways to apply second order correction

This option determins 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 (real): 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.

Default: 0.9999

start_with_resto (string): Tells algorithm 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.

Default: no

valuemeaning
no don't force start in restoration phase
yes force start in restoration phase

s_max (real): Scaling threshold for the NLP error.

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

Default: 100

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

(See Eqn. (19) in the implementation paper.)

Range: [1, ∞]

Default: 2.3

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

(See Eqn. (19) in the implementation paper.)

Range: [1, ∞]

Default: 1.1

tau_min (real): 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 (real): 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).

Default: 10000

theta_min_fact (real): 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 rules 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).

Default: 0.0001

tiny_step_tol (real): 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.

Default: 2.22045e-15

tiny_step_y_tol (real): 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.

Default: 0.01

tol (real): 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.

Default: 1e-08

warm_start_bound_frac (real): same as bound_frac for the regular initializer.

Range: [0, 0.5]

Default: 0.001

warm_start_bound_push (real): same as bound_push for the regular initializer.

Default: 0.001

warm_start_init_point (string): 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.)

Default: no

valuemeaning
no do not use the warm start initialization
yes use the warm start initialization

warm_start_mult_bound_push (real): same as mult_bound_push for the regular initializer.

Default: 0.001

warm_start_mult_init_max (real): Maximum initial value for the equality multipliers.

Range: [-∞, ∞]

Default: 1e+06

warm_start_slack_bound_frac (real): same as slack_bound_frac for the regular initializer.

Range: [0, 0.5]

Default: 0.001

warm_start_slack_bound_push (real): same as slack_bound_push for the regular initializer.

Default: 0.001

watchdog_shortened_iter_trigger (integer): 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.

Default: 10

watchdog_trial_iter_max (integer): 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