$title Simple Nonconvex Bilevel Programming Example (MIRRLEES,12)

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This simple nonconvex example is borrowed from

J. A. Mirrlees. The theory of moral hazard and unobservable bevaviour:
part I. Review of Economic Studies, 66: 3-21, 1999.

Contributor: Jan-H. Jagla, January 2009

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Equations
   outerobj
   innerobj;

Variables
   out
   in;

Positive Variables
   x
   y;

Scalar coef;

outerobj.. out =e= sqr(y-2) + sqr(x-1);

innerobj..  in =e= -y*exp(-sqr(x+1)) - exp(-sqr(x-1));

model outer / outerobj /
      inner / innerobj /
      emp   / outer, inner /;

* Solve the model by reformulating it into a
* Mathematical Programs with Equilibrium Constraints (MPEC)
Equations dLdy;
dLdy.. (1 + x)*exp(-sqr(1 + x))*y + (x - 1)*exp(-sqr(x - 1)) =N= 0;

model bilevel / outer, dLdy.y /;
solve bilevel using mpec minimizing out;
scalar out_l,x_l,y_l;
out_l = out.l;
x_l   = x.l;
y_l   = y.l;

* Do this reformulation automatically using EMP
$echo bilevel y min in x innerobj > "%emp.info%"
solve emp using emp minimizing out;

scalar tol /1e-6/;
abort$(  abs(out_l - out.l)
       + abs(x_l   - x.l)
       + abs(y_l   - y.l) > tol) 'Solutions differ';