cns01.gms : Test how bounds change the 1-variable CNS model sqr(x)=4

**Description**

Single equation square nonlinear system with bounds varying from free over exactly at the solution to a fixed variable. The model without bounds has two solutions, x = 2 and x = -2, and the one that is returned may depend on the initial value. Bounds may of course exclude one of the solutions.

**Small Model of Type : ** CNS

**Category :** GAMS Test library

**Main file : ** cns01.gms

```
$title Test how bounds change the 1-variable CNS model sqr(x)=4 (cns01,SEQ=91)
$ontext
Single equation square nonlinear system with bounds varying
from free over exactly at the solution to a fixed variable.
The model without bounds has two solutions, x = 2 and x = -2,
and the one that is returned may depend on the initial value.
Bounds may of course exclude one of the solutions.
$offtext
maxexecerror = 1;
$if not set TESTTOL $set TESTTOL 1e-6
scalar tol / %TESTTOL% /;
variable x;
equation f;
model m / f /;
f .. x*x =e= 4;
option limrow = 0, limcol = 0;
* Case 1: either x = -2 or x = 2 is OK.
x.l = 8;
solve m using cns;
abort$(m.solvestat <> %solvestat.NormalCompletion% or m.modelstat <> %modelstat.Solved%) 'bad return codes';
abort$(abs(sqr(x.l)-4) > tol) 'x.l should be + or -2',x.l;
* Case 2: either x = -2 or x = 2 is OK.
x.l = -8;
solve m using cns;
abort$(m.solvestat <> %solvestat.NormalCompletion% or m.modelstat <> %modelstat.Solved%) 'bad return codes';
abort$(abs(sqr(x.l)-4) > tol) 'x.l should be + or -2',x.l;
* Case 3: The Jacobian is singular in the initial point, but OK at
* the two solutions. The return will be solver specific.
x.l = 0;
solve m using cns;
abort$(m.solvestat <> %solvestat.NormalCompletion%) 'bad solvestat';
if { (m.modelstat = %modelstat.LocallyInfeasible%),
* locally infeasible, should report a row that way
abort$(m.numinfes < 1) 'wrong .numinfes';
else
abort$(m.modelstat <> %modelstat.Solved%) 'bad modelstat';
abort$(abs(sqr(x.l)-4) > tol) 'x.l should be + or -2',x.l;
};
* Case 4: Must return x = 2 because of the bounds.
x.lo = 1; x.up = 3; x.l = 8;
solve m using cns;
abort$(m.solvestat <> %solvestat.NormalCompletion% or
(m.modelstat <> %modelstat.SolvedUnique% eqv m.modelstat <> %modelstat.Solved%)) 'bad return codes';
abort$(abs(x.l-2) > tol) 'x.l should be 2',x.l;
* Case 5: Must return x = 2 because of the bounds. Note that the
* solution is at the bound, but the bound is not violated.
x.lo = 2; x.up = 3; x.l = 8;
solve m using cns;
abort$(m.solvestat <> %solvestat.NormalCompletion% or
(m.modelstat <> %modelstat.SolvedUnique% eqv m.modelstat <> %modelstat.Solved%)) 'bad return codes';
abort$(abs(x.l-2) > tol) 'x.l should be 2',x.l;
* Case 6a: The model is nonsquare because there are no non-fixed
* variables left but there is one equation. How this is treated
* depends on the holdfixed setting
* If holdfixed is 0 GAMS rejects this model as nonsquare
* - it never gets to the solver
m.holdfixed = 0; x.lo = 2; x.up = 2; x.l = 8; solve m using cns;
abort$(execerror=0) 'previous solve should have given exec errors';
abort$(m.solvestat <> %solvestat.SolveProcessingSkipped% or m.modelstat <> %modelstat.NoSolutionReturned%) 'bad return codes';
execerror = 0;
* Case 6b: If holdfixed = 1 GAMS accepts the model and the solver
* should give a reasonable return. N.B. GAMS' acceptance
* of this model happens somewhat by chance - it will not always
* be the case that setting holdfixed=1 will recover
* squareness. In this case, the number of equations that
* drop out is equal to the number of fixed variables, so we
* stay square
m.holdfixed = 1; x.lo = 2; x.up = 2; x.l = 8; solve m using cns;
abort$(m.solvestat <> %solvestat.NormalCompletion%) 'bad solvestat';
abort$(m.modelstat <> %modelstat.SolvedUnique% and m.modelstat <> %modelstat.Solved%) 'bad modelstat';
* Case 8a: Same checks as 6a and 6b, but the variable is fixed to an
* infeasible point
m.holdfixed = 0; x.lo = 3; x.up = 3; x.l = 8; solve m using cns;
abort$(execerror=0) 'previous solve should have given exec errors';
abort$(m.solvestat <> %solvestat.SolveProcessingSkipped% or m.modelstat <> %modelstat.NoSolutionReturned%) 'bad return codes';
execerror = 0;
* Case 8b: If holdfixed = 1 GAMS will see a square model, but
* the removed row will look like 9 =e= 4. GAMS will
* give an error on model generation.
m.holdfixed = 1; x.lo = 3; x.up = 3; x.l = 8; solve m using cns;
abort$(execerror=0) 'previous solve should have given exec errors';
abort$(m.solvestat <> %solvestat.SolveProcessingSkipped% or m.modelstat <> %modelstat.NoSolutionReturned%) 'bad return codes';
execerror = 0;
```