GAMS — The Solver Manuals - Available Software

126 CONOPT
X.SCALE(I,J,K)$IJK(I,J,K) = expression;
The statement
X.SCALE(I,J,K) = expression;
will generate records for X in the GAMS database for all combinations of I, J, and K for which the expression
is different from 1, i.e. up to 1.e6 records, and apart from spending a lot of time you will very likely run out of
memory. Note that this warning also applies to non-default lower and upper bounds.
7 NLP and DNLP Models
GAMS has two classes of nonlinear model, NLP and DNLP. NLP models are defined as models in which all
functions that appear with endogenous arguments, i.e. arguments that depend on model variables, are smooth
with smooth derivatives. DNLP models can in addition use functions that are smooth but have discontinuous
derivatives. The usual arithmetic operators (+, -, *, /, and **) can appear on both model classes.
The functions that can be used with endogenous arguments in a DNLP model and not in an NLP model are
ABS, MIN, and MAX and as a consequence the indexed operators SMIN and SMAX.
Note that the offending functions can be applied to expressions that only involve constants such as parameters,
var.l, and eq.m. Fixed variables are in principle constants, but GAMS makes its tests based on the functional
form of a model, ignoring numerical parameter values and numerical bound values, and terms involving fixed
variables can therefore not be used with ABS, MIN, or MAX in an NLP model.
The NLP solvers used by GAMS can also be applied to DNLP models. However, it is important to know that the
NLP solvers attempt to solve the DNLP model as if it was an NLP model. The solver uses the derivatives of the
constraints with respect to the variables to guide the search, and it ignores the fact that some of the derivatives
may change discontinuously. There are at the moment no GAMS solvers designed specifically for DNLP models
and no solvers that take into account the discontinuous nature of the derivatives in a DNLP model.
7.1 DNLP Models: What Can Go Wrong
Solvers for NLP Models are all based on making marginal improvements to some initial solution until some
optimality conditions ensure no direction with marginal improvements exist. A point with no marginally improving
direction is called a Local Optimum.
The theory about marginal improvements is based on the assumption that the derivatives of the constraints with
respect to the variables are a good approximations to the marginal changes in some neighborhood around the
current point.
Consider the simple NLP model, min SQR(x), where x is a free variable. The marginal change in the objective
is the derivative of SQR(x) with respect to x, which is 2*x. At x = 0, the marginal change in all directions is
zero and x = 0 is therefore a Local Optimum.
Next consider the simple DNLP model, min ABS(x), where x again is a free variable. The marginal change in
the objective is still the derivative, which is +1 if x > 0 and -1 if x < 0. When x = 0, the derivative depends on
whether we are going to increase or decrease x. Internally in the DNLP solver, we cannot be sure whether the
derivative at 0 will be -1 or +1; it can depend on rounding tolerances. An NLP solver will start in some initial
point, say x = 1, and look at the derivative, here +1. Since the derivative is positive, x is reduced to reduce
the objective. After some iterations, x will be zero or very close to zero. The derivative will be +1 or -1, so the
solver will try to change x. however, even small changes will not lead to a better objective function. The point x
= 0 does not look like a Local Optimum, even though it is a Local Optimum. The result is that the NLP solver
will muddle around for some time and then stop with a message saying something like: ”The solution cannot be
improved, but it does not appear to be optimal.”
In this first case we got the optimal solution so we can just ignore the message. However, consider the following
simple two-dimensional DNLP model: min ABS(x1+x2) + 5*ABS(x1-x2) with x1 and x2 free variables. Start