Version:

emp03.gms : Check correctness of NLP->MCP reform in EMP

**Description**

Test conventions for EMP rewriting as an MCP (aka NLPD). model 1 is a max model, model 2 a min. They are identical except for 1 is max z=f(x), the other is min z=-f(x) and of course the equation duals have opposite sign Key observation: the NLPD option dualVar x f essentially equates f.m with variable x, so we don't compute derivatives wrt. x as we would if x were a primal variable. Since this is so, we assume x has the same sign as f.m would in the NLP model, and we need to do a reformulation that preserves the sign of f.m=x. The model shows how we can reformulate a max model and its equivalent min so that we preserve the sign of the equation duals in the MCP version. Contributor: Steve Dirkse, April 2008

**Small Model of Type : ** GAMS

**Category :** GAMS Test library

**Main file : ** emp03.gms

```
$title 'Check correctness of NLP->MCP reform in EMP' (EMP03,SEQ=388)
$ontext
Test conventions for EMP rewriting as an MCP (aka NLPD).
model 1 is a max model, model 2 a min. They are identical
except for 1 is max z=f(x), the other is min z=-f(x)
and of course the equation duals have opposite sign
Key observation: the NLPD option
dualVar x f
essentially equates f.m with variable x, so we don't compute
derivatives wrt. x as we would if x were a primal variable. Since this
is so, we assume x has the same sign as f.m would in the NLP model, and
we need to do a reformulation that preserves the sign of f.m=x.
The model shows how we can reformulate a max model and its equivalent min
so that we preserve the sign of the equation duals in the MCP version.
Contributor: Steve Dirkse, April 2008
$offtext
variable z;
positive variables x, y;
x.lo = 1e-2;
equations
f1 'objective definition for max model'
f2 'objective definition for min model'
e 'equality'
up 'upper bound'
lo 'lower bound'
;
f1.. z =E= x + y;
f2.. z =E= -x - y;
e.. y =E= sqrt(x);
up.. exp(x) =L= 20;
lo.. y =G= .1 * x;
model nlp1 'max of z=f(x)' / f1, e, up, lo /;
model nlp2 'min of z=-f(x)' / f2, e, up, lo /;
free variables
v1 'perp to e in nlp1'
v2 'perp to e in nlp2';
positive variables
uUp1 'perp to up in nlp1'
uLo2 'perp to lo in nlp2';
negative variable
uLo1 'perp to lo in nlp1'
uUp2 'perp to up in nlp2';
equations
dLdx1, dLdy1
dLdx2, dLdy2
eNeg
upNeg
loNeg
;
dLdx1.. -1 - v1*(0.5/sqrt(x)) + uUp1*exp(x) - uLo1*.1 =N= 0;
dLdy1.. -1 + v1 + uUp1*0 + uLo1 =N= 0;
dLdx2.. -1 + v2*(0.5/sqrt(x)) - uUp2*exp(x) + uLo2*.1 =N= 0;
dLdy2.. -1 - v2 - uUp2*0 - uLo2 =N= 0;
eNeg.. sqrt(x) =E= y;
upNeg.. 20 =G= exp(x);
loNeg.. .1 * x =L= y;
model mcp1 / dLdx1.x, dLdy1.y, eNeg.v1, upNeg.uUp1, loNeg.uLo1 /;
model mcp2 / dLdx2.x, dLdy2.y, e.v2, up.uUp2, lo.uLo2 /;
solve nlp1 using nlp max z;
* now set all the duals:
* the duals for mcp1 use the same sign as nlp1: THAT IS KEY!
v1.l = e.m;
uUp1.l = up.m;
uLo1.l = lo.m;
* the duals for mcp2 flip sign
v2.l = -e.m;
uUp2.l = -up.m;
uLo2.l = -lo.m;
* and of course the duals for nlp2 flip sign also
z.l = -z.l;
f2.m = -f1.m;
e.m = -e.m;
up.m = -up.m;
lo.m = -lo.m;
* these next three solves should not take any iterations
option nlp = minos;
solve nlp2 using nlp min z;
abort$[nlp2.iterusd > 0] 'nlp2 took some iterations';
option mcp = path;
mcp1.iterlim = 0;
solve mcp1 using mcp;
abort$[mcp1.objval > 1e-4] 'mcp1 was not given a solution';
mcp2.iterlim = 0;
solve mcp2 using mcp;
abort$[mcp2.objval > 1e-4] 'mcp2 was not given a solution';
nlp1.optfile = 99;
nlp2.optfile = 99;
$onecho > jams.o99
dict nlpDict.txt
fileName genMCP.gms
$offecho
$echo modeltype mcp > "%emp.info%"
solve nlp2 using emp min z;
abort$[nlp2.iterusd > 0] 'nlp2 took some iterations';
z.l = -z.l;
f1.m = -f2.m;
e.m = -e.m;
up.m = -up.m;
lo.m = -lo.m;
solve nlp1 using emp max z;
abort$[nlp1.iterusd > 0] 'nlp1 took some iterations';
$if not set DUMPMCP $exit
* now we make it really easy to debug this in case the nlp models
* do not solve as expected with EMP. The MCP models below were known
* to work
$onecho > emp03mcp1.gms
* written by GAMS/EMP at 04/12/08 08:20:11
*
Variables x2,x3,u2,u3,u4;
Negative Variables u4;
Positive Variables x3;
Positive Variables u3;
Equations e2,e3,e4,dL_dx2,dL_dx3;
e2.. 0 =E= - sqrt(x2) + x3;
e3.. 20 =G= exp(x2);
e4.. 0 =L= - 0.1*x2 + x3;
dL_dx2.. -1 + ( - 0.5/sqrt(x2))*u2 + (exp(x2))*u3 - 0.1*u4 =N= 0;
dL_dx3.. -1 + u2 + u4 =N= 0;
* set non-default bounds
x2.lo = 0.01;
* set non-default levels
x2.l = 2.99573227355399;
x3.l = 1.73081838260229;
u2.l = 1;
u3.l = 0.0644440342506719;
Model m / e2.u2,e3.u3,e4.u4,dL_dx2.x2,dL_dx3.x3 /;
m.limrow=0; m.limcol=0;
Solve m using MCP;
$offecho
$onecho > emp03mcp2.gms
* written by GAMS/EMP at 04/12/08 08:21:27
*
Variables x2,x3,u2,u3,u4;
Negative Variables u3;
Positive Variables x3;
Positive Variables u4;
Equations e2,e3,e4,dL_dx2,dL_dx3;
e2.. - sqrt(x2) + x3 =E= 0;
e3.. exp(x2) =L= 20;
e4.. - 0.1*x2 + x3 =G= 0;
dL_dx2.. -1 - ( - 0.5/sqrt(x2))*u2 - (exp(x2))*u3 + 0.1*u4 =N= 0;
dL_dx3.. -1 - u2 - u4 =N= 0;
* set non-default bounds
x2.lo = 0.01;
* set non-default levels
x2.l = 2.99573227355399;
x3.l = 1.73081838260229;
u2.l = -1;
u3.l = -0.0644440342506719;
Model m / e2.u2,e3.u3,e4.u4,dL_dx2.x2,dL_dx3.x3 /;
m.limrow=0; m.limcol=0;
Solve m using MCP;
$offecho
```