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Constraints 165
In fact, one application of penalty functions is to acquire feasibility for
inequality constraints. To do so for inequality constraint c i ' use the penalty
constraint function
(5.107)
A little thought will show that this is exactly equivalent to the "satisfied-whenexceeded"
technique discussed in Section 5.5.2. It is seen that the partial
derivatives of h, in (5.107) exist at the boundary of feasibility.
Penalty functions are usually better behaved in unconstrained optimization
than barrier functions. This is usually due to the mechanics of the linear
search process, where the infinite barrier may be overstepped by the necessarily
finite exploratory moves. A review of the example problem and its
treatment by transformation of variables in Section 5.6.1 supports this conclusion.
5.6.4. Mixed Compound FUllCtion for All Constraints. Fiacco and McCormick
(1968) derived the necessary conditions for defining a combined barrier
and penalty function:
with derivatives
M
minF(x)=E(x)+rL _1_+_1 L hUx),
x.' i-I ci(x) Ii k-I
aF aE ~ acJaxj 2.f, ah k
-=--r.::.. +-.::.. hk(x)-.
aXj ax; i-I cr(x) .(r k-I ax;
P
(5.108)
(5.109)
One practical consideration in (5.108) is the choice of the starting value for r.
If it is too small, then the Ci inequality constraint barriers will be too far away
and steep, so that the h, penalty functions wil1 tend to dominate the objective
E(x). Difficulties of the opposite nature exist if the initial r is too large. There
are fairly sophisticated means for selecting the initial r value, but one way that
at least leaves the objective E(x) somewhat in control has been satisfactory.
The value of E(x) and of each summation in (5.108) is obtained for the
contemplated starting point in variable (x) space. Then the first r value is
chosen so that the absolute value of barrier and penalty contributions is just
10% of the E(x) contribution to F(x,r). This procedure requires the solution of
a real quadratic equation.
Fiacco and McCormick (1968) also show why and how the Richardson
extrapolation to the limit operates. Using this extrapolation for all variables
often places the solution inside unfeasible regions. In short, there are some
programming complexities to be overcome in applying the Richardson extrapolation
to barrier, penalty, and mixed functions. The good news is that
personal computer users operate in the loop with program execution. The
complicated program features required in a timeshare environment to avoid