$title Simple bilevel model, both inner and outer problems have constraints (EMPBP03,SEQ=422)

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This model compares the results obtained by EMP with the ones obtained using the
corresponding hand-crafted model. Both, the outer minimization problem and the
inner minimization problem contains constraints

Contributor: Jan-H. Jagla, January 2009

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$if not set nlpsolver $set nlpsolver conopt

*-------------------------------------------------------------------------------
*Prepare reporting

Parameter rep comparison;
sets
   enames     / outerobj,outereq1,innerobj,innereq1 /
   vnames     / out,x1,x2,in,y1,y2,y3 /
   evnames    / set.enames,set.vnames /
   suf        / l /;

$gdxOut "%gams.scrdir%repsets.%gams.scrext%"
$unLoad evnames suf
$gdxOut

$onEcho > "%gams.scrdir%repmaker.%gams.scrext%"
set evnames, suf;
$gdxIn "%gams.scrdir%repsets.%gams.scrext%"
$load evnames suf

file fx /"%gams.scrdir%report.%gams.scrext%"/; put fx;
loop((evnames,suf),
$escape =
  put "rep('" evnames.te(evnames) "','" suf.te(suf) "','%=1')"
  put " = " evnames.te(evnames) "." suf.te(suf) ";" /;
$escape %
);
$offEcho

$call gams "%gams.scrdir%repmaker.%gams.scrext%" lo=0
*-------------------------------------------------------------------------------

Equations
   outerobj
   outereq1
   innerobj
   innereq1;

Variables
   out
   in;

Positive Variables
   x1
   x2
   y1
   y2
   y3;

outerobj.. out =e= sqr(y1) + sqr(y3) - y1*y3 - 4*y2 - 7*x1 + 4*x2;

outereq1..  x1 + x2 =l= 1;

innerobj.. in =e= sqr(y1) + 0.5*sqr(y2) + 0.5*sqr(y3) + y1*y2 + (1-3*x1)*y1 + (1+x2)*y2;

innereq1.. 2*y1 + y2 - y3 + x1 - 2*x2 + 2 =l= 0;

model outer   / outerobj, outereq1 /
      inner   / innerobj, innereq1 /
      bilevel / outer, inner       /;

$echo bilevel x1 x2 min in * innerobj innereq1 > "%emp.info%"

option dnlp=%nlpsolver%;

solve  bilevel using emp mininizing out;
abort$(bilevel.solvestat <> %solveStat.normalCompletion%) 'EMP solver status <> 1';
$batInclude "%gams.scrdir%report.%gams.scrext%" emp

*Verify solution

*KKT conditions for inner minimization problem
Negative Variables  u2;
Equations  dLdy1,dLdy2,dLdy3;
dLdy1.. ( - (2*y1 + y2 + 1 - 3*x1))/(-1) - u2*2 =N= 0;
dLdy2.. ( - (y2 + y1 + 1 + x2))/(-1) - u2 =N= 0;
dLdy3.. ( - y3)/(-1) + u2 =N= 0;

Model manual / outer, innereq1.u2, dLdy1.y1, dLdy2.y2, dLdy3.y3 /;

solve manual using mpec minimizing out;
abort$(manual.solvestat <> %solveStat.normalCompletion%) 'Hand-made model solver status <> 1';
$batInclude "%gams.scrdir%report.%gams.scrext%" manual

*-------------------------------------------------------------------------------
*Reporting Differences

rep(evnames,suf,'diff') = abs(rep(evnames,suf,'emp') - rep(evnames,suf,'manual'));
rep(evnames,suf,'diff')$(rep(evnames,suf,'diff') < 1e-5) = 0;
display rep;
abort$sum((evnames,suf),rep(evnames,suf,'diff')) 'Solutions differ';