Table of Contents
Quasi-variational inequalities are a generalization of the variational inequality model: in a VI, the feasible set is fixed, while the QVI allows the feasible set to vary with or be a function of the variables in the model. To avoid repetition, we assume you are already familiar with the theory and notation for VI models. In this section, we present a mathematical formulation of QVI, give an example of how QVI can be modeled with GAMS EMP, and introduce the EMP annotations specific to QVI.
Quasi-Variational Inequalities: Mathematical Formulation
For a given continuous function
where
If the point-to-set mapping
Quasi-Variational Inequalities with EMP: A Simple Example
Consider the following simple two dimensional linear example. Let
This
set i / 1*2 /;
alias(i,j);
table A(i,j)
1 2
1 2 [8/3]
2 [5/4] 2 ;
parameters
b(i) / 1 [100/3], 2 22.5 /
Cy(i,j) / 1.1 1, 2.2 1 /
Cx(i,j) / 1.2 1, 2.1 1 /
rhs(i) / 1 15, 2 20 /
;
positive variables
y(j) 'variable of interest, aka decision variable'
x(j) 'parameter variable shadowing y'
;
y.up(j) = 11; x.up(j) = 11;
equations
F(i) 'FOC for agent optimization models'
g(i) 'define feasible set K(x) for QVI'
;
F(i).. sum{j, A(i,j)*y(j)} - b(i) =N= 0;
g(i).. sum{j, Cy(i,j)*y(j)} + sum{j, Cx(i,j)*x(j)} =L= rhs(i);
model m / F, g /;
file annotations / '%emp.info%' /;
putclose annotations 'qvi F y x g' ;
solve m using emp;
Observe that the function qvi
indicates that the model is a QVI, that the QVI function F
is matched to the variable-of-interest y
with its parameter variable x
, and that the constraints g
define the mapping
Since QVI problems have no objective, the short form of the solve statement is used.
The solver JAMS will reformulate the QVI as an MCP and pass this on to an MCP subsolver. The EMP Summary produced by JAMS will contain the following lines:
--- EMP Summary ... VI Functions = 2 QVI Parameters = 2 ...
This output reflects the fact that there were two VI functions in the model above, one for each member of the set i
, and that each of the variables matched to these functions was shadowed by a parameter variable.
Note that there are two QVI models in the GAMS EMP Library, the models [SIMPLEQVI1] and [SIMPLEQVI2] (an expanded version of the example shown above).
EMP Syntax for Quasi-Variational Inequalities
The general syntax of the EMP annotations used to specify quasi-variational inequalities is as follows:
QVI {0 var [ parameterVar] | [-] equ var [ parameterVar] } { [-] equ}
The EMP keyword QVI
indicates that this is a quasi-variational inequality specification. All variables and equations included in a QVI must be listed explicitly. First we have the VI functions, their matching variables, and (optionally) the parameter variables shadowing these variables. Note that in a QVI there are no preceding variables as we have in a VI spec:: instead, the implied match to the zero function is indicated by the digit 0
appearing where an equation symbol would otherwise appear. After the function/variable pairs have been listed, the trailing equations (i.e. the equations/constraints defining the mapping