GAMS — The Solver Manuals - Available Software

122 CONOPT
by the reduced complexity. If the model is solved with CONOPT3 using explicit second derivatives then simple
expressions will result in sparser second derivatives that are both faster to compute and to use.
The reduction in complexity can be significant if an intermediate expression is linear.
fragment:
VARIABLE X(I), Y;
EQUATION YDEF;
YDEF .. Y =E= 1 / SUM(I, X(I) );
should be written as
VARIABLE X(I), XSUM, Y;
EQUATION XSUMDEF, YDEF;
XSUMDEF .. XSUM =E= SUM(I, X(I) );
YDEF .. Y =E= 1 / XSUM;
XSUM.LO = 1.E-2;
The following model
for three reasons. First, because the number of nonlinear derivatives is reduced in number and complexity.
Second, because the lower bound on the intermediate result will bound the search away from the singularity at
XSUM = 0. And third, because the matrix of second derivatives for the last model only depend on XSUM while
it depends on all X in the first model.
The last example shows an added potential saving by expanding functions of linear expressions. A constraint
depends in a nonlinear fashion on the accumulated investments, INV, like
CON(I) .. f( SUM( J$(ORD(J) LE ORD(I)), INV(J) ) ) =L= B(I);
A new intermediate variable, CAP(I), that is equal to the content of the SUM can be defined recursively with
the constraints
CDEF(I) .. CAP(I) =E= INV(I) + CAP(I-1);
and the original constraints become
CON(I) .. f( CAP(I) ) =L= B(I);
The reformulated model has N additional variables and N additional linear constraints. In return, the original
N complex nonlinear constraints have been changed into N simpler nonlinear constraints. And the number of
Jacobian elements, that has a direct influence on much of the computational work both in GAMS and in CONOPT,
has been reduced from N*(N+1)/2 nonlinear elements to 3*N-1 linear elements and only N nonlinear element. If
f is an invertable increasing function you may even rewrite the last constraint as a simple bound:
CAP.LO(I) = finv(B(I));
Some NLP solvers encourage you to move as many nonlinearities as possible into the objective which may make
the objective very complex. This is neither recommended nor necessary with CONOPT. A special pre-processing
step (discussed in section A4 in Appendix A) will aggregate parts of the model if it is useful for CONOPT without
increasing the complexity in GAMS.
6.4 Equalities vs. Inequalities
A resource constraint or a production function is often modeled as an inequality constraint in an optimization
model; the optimization algorithm will search over the space of feasible solutions, and if the constraint turns