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empvi05.gms : Test for EMP's treatment of a odd variables

**Description**

This model verifies that EMP treats the variable jj correctly. It only occurs linearly in the objective of mNLP. Considering the corresponding MCP: GAMS' mcp check will fail if a variable that is mapped to a constraint does not show up in the model. One workaround to keep the variable in the model is using eps*jj (see model mMCP). The MCP can be generated and solved through EMP automatically. EMP will preprocess the model and omit jj and dldJJ in generated submodel. Hence, the workaround using eps*jj is not reuired but EMP needs to make sure that jj.l/m are set to correct values (see model mNLP). Consider the corresponding VI: Similar to the MCP we need to include jj using eps*jj (see model mVI) Finally we write down a cognate VI (not the same!) which is a simple example of a VI that cannot be formulated as an opimization problem. This model is a variant of the zerofunc model from the GAMS EMP Library. Contributor: Jan-H. Jagla, March 2009

**Small Model of Type : ** GAMS

**Category :** GAMS Test library

**Main file : ** empvi05.gms

```
$title Test for EMP's treatment of an odd variable (EMPVI05,SEQ=435)
$ontext
This model verifies that EMP treats the variable jj correctly. It only occurs
linearly in the objective of mNLP.
Considering the corresponding MCP: GAMS' mcp check will fail if a variable that
is mapped to a constraint does not show up in the model. One workaround to keep
the variable in the model is using eps*jj (see model mMCP).
The MCP can be generated and solved through EMP automatically. EMP will
preprocess the model and omit jj and dldJJ in generated submodel. Hence, the
workaround using eps*jj is not reuired but EMP needs to make sure that jj.l/m
are set to correct values (see model mNLP).
Consider the corresponding VI: Similar to the MCP we need to include jj using
eps*jj (see model mVI)
Finally we write down a cognate VI (not the same!) which is a simple example of
a VI that cannot be formulated as an opimization problem.
This model is a variant of the zerofunc model from the GAMS EMP Library.
Contributor: Jan-H. Jagla, March 2009
$offtext
file myinfo / '%emp.info%' /;
**** A simple NLP
free variables y, z, jj;
free variable obj;
equations objDef, gCons;
jj.lo=3;
objDef.. obj =E= power(y,3) / 3 + jj;
gCons.. y =G= sqr(z);
model mNLP / objDef, gCons /;
solve mNLP min obj using nlp;
abort$[jj.l <> jj.lo] 'mNLP: jj not at lower bound';
**** The corresponding MCP
positive variable u 'explicit multiplier for MCP';
equations dLdy, dLdz, dLdjj, dLdu;
u.l = gCons.m;
dLdy.. sqr(y) - u =N= 0;
dLdz.. 0 - u*2*z =N= 0;
dLdjj.. 1 + eps*jj - 0 =N= 0;
dLdu.. y - sqr(z) =N= 0;
model mMCP / dLdy.y, dLdz.z, dLdjj.jj, dLdu.u/;
solve mMCP using mcp;
abort$[mMCP.modelStat <> %MODELSTAT.OPTIMAL% ] 'mMCP: Model status not 1';
abort$[mMCP.solveStat <> %SOLVESTAT.NORMAL COMPLETION% ] 'mMCP: Solver status not 1';
abort$[mMCP.objval > 1e-6 ] 'mMCP: Input for model was not optimal';
abort$[mMCP.iterusd > 0 ] 'mMCP: Iterations used';
abort$[jj.l <> jj.lo ] 'mMCP: jj not at lower bound';
**** Create and solve the corresponding MCP through EMP
* tell EMP to solve this as an MCP
putclose myinfo 'ModelType MCP';
solve mNLP min obj using emp;
abort$[mNLP.modelStat <> %MODELSTAT.LOCALLY OPTIMAL% ] 'mNLP: Model status not 2';
abort$[mNLP.solveStat <> %SOLVESTAT.NORMAL COMPLETION% ] 'mNLP: Solver status not 1';
abort$[mNLP.objval <> obj.l ] 'mNLP: ObjVal and obj.l differ';
abort$[mNLP.iterusd > 0 ] 'mNLP: Iterations used';
abort$[jj.l <> jj.lo ] 'mNLP: jj not at lower bound';
**** Create and solve the corresponding VI through EMP
equation F_y, F_jj;
F_y.. sqr(y) =N= 0;
F_jj.. 1 + eps*jj =N= 0;
* define the complementary pairs for vi
put myinfo 'vi' /;
putclose 'F_y y F_jj jj';
model mVI / F_y, F_jj, gCons /;
solve mVI using emp;
abort$[mVI.modelStat <> %MODELSTAT.LOCALLY OPTIMAL% ] 'mVI: Model status not 1';
abort$[mVI.solveStat <> %SOLVESTAT.NORMAL COMPLETION% ] 'mVI: Solver status not 1';
abort$[mVI.objval > 1e-6 ] 'mVI: Input for model was not optimal';
abort$[mVI.iterusd > 0 ] 'mVI: Iterations used';
abort$[jj.l <> jj.lo ] 'mVI: jj not at lower bound';
********************************************************************************
**** Now use EMP VI with a VI Function that can NOT be derived from an
**** optimzation problem
****
**** Considering F_y only the corresponding NLP objective would look like this:
**** objDef.. obj =E= power(y,3) / 3 + y*jj;
**** However, in this case we would also end up with F_jj:
**** F_jj.. y =N= 0;
equation mod_F_y;
mod_F_y.. sqr(y) + jj =N= 0;
* define the complementary pairs for vi
put myinfo 'vi' /;
putclose 'mod_F_y y';
model mVI_pure / mod_F_y, gCons /;
jj.up = jj.lo;
gCons.m = 3;
solve mVI_pure using emp;
abort$[mVI_pure.modelStat <> %MODELSTAT.LOCALLY OPTIMAL% ] 'mVI_pure: Model status not 1';
abort$[mVI_pure.solveStat <> %SOLVESTAT.NORMAL COMPLETION% ] 'mVI_pure: Solver status not 1';
abort$[mVI_pure.objval > 1e-6 ] 'mVI_pure: Input for model was not optimal';
abort$[mVI_pure.iterusd > 0 ] 'mVI pure: Iterations used';
abort$[jj.l <> jj.lo ] 'mVI_pure: jj not at lower bound';
```