$title Nonlinear Test Problem - Multiple Solutions (MHW4DXX,SEQ=295)

$onText
This popular test problem has several local solutions, local minima
with obj value of 27.87190522 and 52.90257967, and the global minimum
with an obj value of 0.02931083.  Additional programming is done to
test for the correct solution.

We call BARON with the appropriate option to get both solution with one solve.

Wright, M H, Numerical Methods for Nonlinearly Constraint Optimization.
PhD thesis, Stanford University, 1976.

Keywords: nonlinear programming, mathematics, global optimization
$offText

Set i / 1*5 /;

Variable m, x(i);

Equation funct, eq1, eq2, eq3;

funct..
   m =e= sqr(x('1') - 1) + sqr(x('1') - x('2')) + power(x('2') - x('3'),3)
      +  power(x('3') - x('4'),4) + power(x('4') - x('5'),4);

eq1.. x('1') + sqr(x('2')) + power(x('3'),3) =e= 3*sqrt(2) + 2;

eq2.. x('2') - sqr(x('3')) + x('4')          =e= 2*sqrt(2) - 2;

eq3.. x('1')*x('5') =e= 2;

Model wright / all /;

x.l('1') = -1;
x.l('2') =  2;
x.l('3') =  1;
x.l('4') = -2;
x.l('5') = -2;

$ifI not "%gams.nlp%"==BARON $exit

option limCol = 0, limRow = 0;

* there are at least three locally optimal solutions:
* 52.90257967
* 27.87190522
*  0.02931083 global

$onEcho > baron.opt
numSol  3
iSolTol 1
gdxOut  mhw4dxx
$offEcho

wright.optFile =   1;
wright.optCa   = 100;
solve wright using nlp minimizing m;

Set
   solset / mhw4dxx1, mhw4dxx2, mhw4dxx3 /
   asolset(solset) 'solution found by solver';

Variable mall(solset), xall(solset,i);

execute      'gdxmerge mhw4dxx*.gdx > "%gams.scrdir%merge.%gams.scrext%"';
execute_load 'merged.gdx', asolset = merged_set_1, mall = m, xall = x;

option  decimals = 8;
display mall.l, xall.l;

$eolCom //

Scalar tol / 1e-4 /;

* Check feasibility of solutions returned by BARON
loop(asolset,
   m.l    = mall.l(asolset);
   x.l(i) = xall.l(asolset,i);
   abort$(abs(m.l - sqr(x.l('1') - 1) - sqr(x.l('1') - x.l('2')) - power(x.l('2') - x.l('3'),3)
                  - power(x.l('3') - x.l('4'),4) - power(x.l('4') - x.l('5'),4)) > tol) 'funct is bad';
   abort$(abs(x.l('1') + sqr(x.l('2')) + power(x.l('3'),3) - 3*sqrt(2) - 2) > tol) 'e1 is bad';
   abort$(abs(x.l('2') - sqr(x.l('3')) + x.l('4') - 2*sqrt(2) + 2) > tol) 'e2 is bad';
   abort$(abs(x.l('1')*x.l('5') - 2) > tol) 'e3 is bad';
);