multmpec.gms : Educational bilevel model with VI followers

**Description**

This model demonstrate how to use EMP for a bilevel model with multiple inner variational inequality followers. The actual model to solve is: min_{u,v,w,z} z s.t. exp(z) + w = 2, z >= 1 (u,v) solves VI( [v+w+z-1; u-log(v)], {(u,v) | u >= 0, v >= 0 } ) w solves VI( w+z+3, { w | w free } ) Note that the two VI's (due to the definitional sets) correspond respectively to a complementarity problem: 0 <= u \perp v + w + z - 1 >= 0 0 <= v \perp u - log(v) >= 0 and a linear equation: w + z + 3 = 0 The starting value for v is needed to protect the evaluation of log(v). Contributor: Michael Ferris and Jan-H. Jagla, December 2009

**Small Model of Type : ** BP

**Category :** GAMS EMP library

**Main file :** multmpec.gms

```
$title Educational bilevel model with VI followers (MULTMPEC,SEQ=25)
$ontext
This model demonstrate how to use EMP for a bilevel model with multiple
inner variational inequality followers.
The actual model to solve is:
min_{u,v,w,z} z
s.t. exp(z) + w = 2, z >= 1
(u,v) solves VI( [v+w+z-1; u-log(v)], {(u,v) | u >= 0, v >= 0 } )
w solves VI( w+z+3, { w | w free } )
Note that the two VI's (due to the definitional sets) correspond respectively
to a complementarity problem:
0 <= u \perp v + w + z - 1 >= 0
0 <= v \perp u - log(v) >= 0
and a linear equation:
w + z + 3 = 0
The starting value for v is needed to protect the evaluation of log(v).
Contributor: Michael Ferris and Jan-H. Jagla, December 2009
$offtext
positive variable u;
variables v, w, z;
equations f1, f2, f3, h;
f1.. v + w + z =n= 1;
f2.. u =n= log(v);
f3.. w + z =n= -3;
h.. exp(z) + w =e= 2;
v.lo = 0;
v.l = 1;
z.lo = 1;
model mpec /all/;
$onecho > %emp.info%
bilevel
vi f1 u
f2 v
vi f3 w
$offecho
solve mpec using emp min z;
```