GAMS — The Solver Manuals - Available Software

CONOPT 131
Although Conic constraints are convex and therefore usually are considered nice they have one bad property, seen
from an NLP perspective: The derivatives and/or the dual variables are not well defined at the origin, y(i) = 0,
because certain constraint qualifications do not hold. With GCForm = 0 and x = 0 the constraint is effectively
sum(i, sqr(y(i) ) ) =E= 0 that only has the solution y(i) = 0. Since all derivatives are zero the constraint
seems to vanish, but if it still is binding the dual variable will go towards infinity, causing all kinds of numerical
problems. With GCForm = 1 the first derivatives do not vanish in the same way. The y-part of the derivative
vector is a unit vector, but its direction becomes undefined at y(i) = 0 and the second derivatives goes towards
infinity.
The CONOPT option GCPtb1 is a perturbation used to make the functions smooth around the origin. The
default value is 1.e-6 and there is a lower bound of 1.e-12. The GCPtb1 smoothing increases the value of the right
hand side, making the constraint tighter around the origin with a diminishing effect for larger y-values. GCPtb2
is used to control the location of the largest effect of the perturbation. With GCPtb2 = 0 (the default value) the
constraint is tightened everywhere with the largest change of sqrt(GCPtb1) around the origin. With GCPtb2 =
sqrt(GCPtb1) the constraint will go through the origin but will be relaxed with up to GCPtb2 far from the origin.
For many convex model GCPtb2 = 0 will be a good value. However, models in which it is important the x = 0
is feasible, e.g. models with binary variables and constraints of the form x =L= C*bin GCPtb2 must be defined
as sqrt(GCPtb1).
The recommendation for selecting the various Conic options is therefore:
• If you expect the solution to be away from the origin then choose the default GCForm = 0.
• If the origin is a relevant point choose GCForm = 1. If the model is difficult to solve you may try to solve
it first with a large value of GCPtb1, e.g. 1.e-2, and then re-solve it once or twice each time with a smaller
value.
• If you have selected GCForm = 1, select GCPtb2 = sqrt(GCPtb1) if it is essential that x = 0 is feasible.
Otherwise select the default GCPtb2 = 0.
The variables appearing in the Cone constraints are initialized as any other NLP variables, i.e. they are initialized
to zero, projected on the bounds if appropriate, unless the modeler has selected other values. Since Cone
constraints often behave poorly when y(i) = 0 it is a good idea to assign sensible non-zero values to y(i). The
x-values are less critical, but it is also good to assign x-values that are large enough to make the constraints
feasible. If you use GCForm = 1, remember that the definition of feasibility includes the perturbations.
9 APPENDIX A: Algorithmic Information
The objective of this Appendix is to give technically oriented users some understanding of what CONOPT is
doing so they can get more information out of the iteration log. This information can be used to prevent
or circumvent algorithmic difficulties or to make informed guesses about which options to experiment with to
improve CONOPT’s performance on particular model classes.
A1
Overview of GAMS/CONOPT
GAMS/CONOPT is a GRG-based algorithm specifically designed for large nonlinear programming problems
expressed in the following form
min or max f(x) (1)
subject to g(x) = b (2)
lo < x < up (3)
where x is the vector of optimization variables, lo and up are vectors of lower and upper bounds, some of which
may be minus or plus infinity, b is a vector of right hand sides, and f and g are differentiable nonlinear functions
that define the model. n will in the following denote the number of variables and m the number of equations. (2)
will be referred to as the (general) constraints and (3) as the bounds.