GAMS — The Solver Manuals - Available Software

148 CONOPT
X2 appearing in
E2: Pivot too small for triangular model. Value=0.000E+00
** Infeasible solution. The equations were assumed to be
recursive but they are not. A pivot element is too small.
However, the uniqueness of X2 may not be relevant if the solution just is going to be used as an initial point for
a second model. The option ”lsismp = t” (for Logical Switch: Ignore SMall Pivots) will allow zero pivots as
long as the corresponding equation is feasible for the given initial values.
A13.2 Constrained Nonlinear System or Square Systems of Equations
There is a special model class in GAMS called CNS - Constrained Nonlinear System. A constrained nonlinear
system is a square system of equations, i.e. a model in which the number of non-fixed variables is equal to the
number of constraints. Currently, CONOPT2 and PATH are the only solvers for this model class. A CNS model
can be solved with a solve statement like
SOLVE USING CNS;
without an objective term. In some cases it may be convenient to solve aCNS model with a standard solve
statement combined with an options file that has the statement ”lssqrs = t”. In the latter case, CONOPT will
check that the number of non-fixed variables is equal to the number of constraints. In either case, CONOPT
will attempt to solve the constraints with respect tothe non-fixed variables using Newton’s method. The solution
process will stop with an error message and the current intermediate infeasible solution will be returned if the
Jacobian to be inverted is singular, or if one of the non-fixed variables tries to move outside their bounds.
Slacks in inequalities are counted as non-fixed variables which effectively means that inequalities should not be
binding. Bounds on the variables are allowed, especially to prevent function evaluation errors for functions that
only are defined for some arguments, but the bounds should not be binding in the final solution.
The solution returned to GAMS will in all cases have marginal values equal to 0 or EPS, both for the variables
and the constraints.
The termination messages for CNS models are different from the termination messages for optimization models.
The message you hope for is
** Feasible solution to a square system.
that usually will be combined with model status 16-Solved. If CONOPT in special cases can guarantie that the
solution is unique, for example if the model is linear, then the model status will be 15-Solved Unique.
There are two potential error termination messages related to CNS models. A model with the following two
constraints
e1 .. x1 + x2 =e= 1;
e2 .. 2*x1 + 2*x2 =e= 2;
will result in the message
** Error in Square System: Pivot too small.
e2: Pivot too small.
x1: Pivot too small.
”Pivot too small” means that the set of constraints is linearly dependent and there cannot be a unique solution
to the model. The message points to one variable and one constraint. However, this just indicates that the
linearly dependent set of constraints and variables include the constraint and variable mentioned. The offending
constraint and variable will also be labeled ’DEPND’ for linearly dependent in the equation listing. The error