GAMS — The Solver Manuals - Available Software

130 CONOPT
the NLP solver will compute constraint values and derivatives in various points within the bounds defined by
the modeler. If these calculations result in undefined intermediate or final values, a function evaluation error is
reported, an error counter is incremented, and the point is flagged as a bad point. The following action will then
depend on the solver. The solver may try to continue, but only if the modeler has allowed it with an ”Option
Domlim = xxx”. The problem of detecting discontinuities is changed from a structural test at the GAMS model
generation stage to a dynamic test during the solution process.
You may have a perfectly nice model in which intermediate terms become undefined. The composite function
SQRT( POWER(x,3) ) is mathematically well defined around x = 0, but the computation will involve the
derivative of SQRT at zero, that is undefined. It is the modeler’s responsibility to write expressions in a way that
avoids undefined intermediate terms in the function and derivatives computations. In this case, you may either
add a small strictly positive lower bound on x or rewrite the function as x**1.5.
8 Conic Constraints with GAMS/CONOPT
Certain types of conic constraints can be formulated in GAMS as described in the GAMS/MOSEK user’s guide.
The GAMS/CONOPT interface translates these constraints into nonlinear constraints and treats them as described
in this note.
The quadratic cone is described in GAMS as
Qcone.. x =C= sum(i, y(i) );
and it represents the convex nonlinear constraint
x > sqrt( sum(i, sqr( y(i) ) ) ).
The rotated quadratic (or hyperbolic) cone is described in GAMS as
Hcone.. x1 + x2 =C= sum(i, y(i) );
and it represents the convex nonlinear constraint
sqrt(2*x1*x2) > sqrt( sum(i, sqr( y(i) ) ) ) with x1 > 0 and x2 > 0.
The cones are in GAMS/CONOPT implemented using one of two mathematical forms. The mathematical form
is selected from the CONOPT option GCForm as follows:
GCForm = 0 (the default value):
QCone.. sqr(x) =G= sum(i, sqr( y(i) ) );
Hcone.. 2*x1*x2 =G= sum(i, sqr( y(i) ) );
GCForm = 1:
QCone.. x+GCPtb2 =G= sqrt( GCPtb1+sum(i, sqr( y(i) ) ) );
Hcone.. Sqrt( GCPtb1 + 2*x1*x2 ) =G= Sqrt( GCptb1+sum(i, sqr( y(i) ) ) );
where GCPtb1 and GCPtb2 are perturbation parameters (explained below).
The advantages and disadvantages of the two formulations are as follows: With GCForm = 0 all functions are
quadratic with a sparse Hessian and bounded second derivatives. However, function values grow with sqr(x) and
first derivatives grow with x and CONOPT’s automatic scaling methods will sometimes have problems selecting
good scaling factors for these equations. With GCForm = 1 the functions are more complicated with dense Hessian
matrices. However, the function values grow linearly with x and the first derivatives are unit vectors which usually
gives a nicely scaled model.