Version:

empvi04.gms : Test for EMP Variational Inequalities

**Description**

VI is to find x in X: F(x) (y - x) >= 0, for all y in X X is a closed convex set, e.g. X = { x >= 0 | g(x) >= 0 } This can be reformulated as an MCP: 0 <= F(x) - \grad g(x) u \perp x >= 0 0 <= g(x) \perp u >= 0 In what follows, the model reform is the MCP that EMP should Contributor: Steven Dirkse and Jan-H. Jagla , January 2009

**Small Model of Type : ** GAMS

**Category :** GAMS Test library

**Main file : ** empvi04.gms

```
$title Test for EMP Variational Inequalities (EMPVI04,SEQ=426)
$onText
VI is to find x in X:
F(x) (y - x) >= 0, for all y in X
X is a closed convex set, e.g. X = { x >= 0 | g(x) >= 0 }
This can be reformulated as an MCP:
0 <= F(x) - \grad g(x) u \perp x >= 0
0 <= g(x) \perp u >= 0
In what follows, the model reform is the MCP that EMP should
Contributor: Steven Dirkse and Jan-H. Jagla , January 2009
$offText
sets
I / i1, i2 /
J / j1 * j3 /
;
table A(I,J)
j1 j2 j3
i1 1 1
i2 1 1 ;
parameter b(I) /
i1 6
i2 9
/;
positive variables
x(J) 'primal vars, perp to f(J)'
u(I) 'auxiliary vars, perp to g(I)'
;
equations
F(J)
F_aux(J)
g(I)
;
F(J).. 2 * x(J) =N= 0 ;
F_aux(J).. 2 * x(J) + sum{I, -A(I,J)*u(I)} =N= 0 ;
g(I).. sum {j, A(I,J)*x(J)} =g= b(I) ;
* VI re-formulated as MCP
model reform / F_aux.x, g.u/;
solve reform using mcp;
model niceVI / F, g/;
file fx / "%emp.info%" /;
putclose fx 'vi F x';
niceVI.iterlim = 0;
solve niceVI using emp;
abort$[niceVI.objval > 1e-6] 'Input for model vi should be optimal, was not';
```