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128 CONOPT
delta. The error is reduced to approximately SQR(delta)/2 for f(x) = 1. The second derivative is 1/delta at f(x)
= 0 (excluding terms related to the second derivative of f(x)). A delta value between 1.e-3 and 1.e-4 should in
most cases be appropriate. It is possible to use a larger value in an initial optimization, reduce it and solve the
model again. You should note, that if you reduce delta below 1.e-4 then large second order terms might lead to
slow convergence or even prevent convergence.
The approximation shown above has its largest error when f(x) = 0 and smaller errors when f(x) is far from
zero. If it is important to get accurate values of ABS exactly when f(x) = 0, then you may use the alternative
approximation
SQRT( SQR(f(x)) + SQR(delta) ) - delta
instead. The only difference is the constant term. The error is zero when f(x) is zero and the error grows to -delta
when f(x) is far from zero.
Some theoretical work uses the Huber, H(*), function as an approximation for ABS. The Huber function is
defined as
H(x) = x for x > delta,
H(x) = -x for x < -delta and
H(x) = SQR(x)/2/delta + delta/2 for -delta < x < delta.
Although the Huber function has some nice properties, it is for example accurate when ABS(x) > delta, it is not
so useful for GAMS work because it is defined with different formulae for the three pieces.
A smooth GAMS approximation for MAX(f(x),g(y)) is
( f(x) + g(y) + SQRT( SQR(f(x)-g(y)) + SQR(delta) ) )/2
where delta again is a small scalar. The approximation error is delta/2 when f(x) = g(y) and decreases with the
difference between the two terms. As before, you may subtract a constant term to shift the approximation error
from the area f(x) = g(y) to areas where the difference is large. The resulting approximation becomes
( f(x) + g(y) + SQRT( SQR(f(x)-g(y)) + SQR(delta) ) - delta )/2
Similar smooth GAMS approximations for MIN(f(x),g(y)) are
and
( f(x) + g(y) - SQRT( SQR(f(x)-g(y)) + SQR(delta) ) )/2
( f(x) + g(y) - SQRT( SQR(f(x)-g(y)) + SQR(delta) ) + delta )/2
Appropriate delta values are the same as for the ABS approximation: in the range from 1.e-2 to 1.e-4.
It appears that there are no simple symmetric extensions for MAX and MIN of three or more arguments or for
indexed SMAX and SMIN.
7.4 Are DNLP Models Always Non-smooth
A DNLP model is defined as a model that has an equation with an ABS, MAX, or MIN function with endogenous
arguments. The non-smooth properties of DNLP models are derived from the non-smooth properties of these
functions through the use of the chain rule. However, composite expressions involving ABS, MAX, or MIN can
in some cases have smooth derivatives and the model can therefore in some cases be smooth.
One example of a smooth expression involving an ABS function is common in water systems modeling. The
pressure loss over a pipe, dH, is proportional to the flow, Q, to some power, P. P is usually around +2. The sign
of the loss depend on the direction of the flow so dH is positive if Q is positive and negative if Q is negative.