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-70.
Constrainis 163
315-325). For small computers, systems of constraints defined by (5.101) may
be treated by the following barrier technique.
A barrier function for generally nonlinear constraints in vector cis:
M
minQ(x)=E(x)+r2; _(1);
x i=l c j X
(5.102)
The nature of a barrier function is seen by considering the following objective
function:
minE(x)=4x,+x,; xispositive. (5.103)
This function is shown in Figure 5.32. Clearly, the constrained optimum is at
the origin, as indicated. The barrier function corresponding to· (5.102) has
already been written; it is (5.96), for which figure 5.30 applies. Note that the
value r= I produced a minimum at x=(0.5, ll. The barrier is created by the
infinite contours of r/x, and rlx, or, in general, ric, for the ith constraint
approaching zero, the edge of its feasible region.
The barrier function is employed in asequence of unconstrained minimiza~
tions, each for a smaller value of parameter r in (5.102). (It can be shown
analytically that the limit at r = 0 exists.) An expression for these minima can
he written for barrier function (5.96) by setting its partial derivatives equal to
zero:
N =4- I- =0
aXl x~'
N=l-I-=O.
aX2 xi
(5.104)
(5.105)
Any particular minimum occurs at Xl = If12 and x,= If. Eliminating r shows
x,
Feasible region:
region:
XI #' 0
X 2 #' a
Optimum
(0,0)
x,
E(x) '" 0 1 2 3 4 5 6 7 8
Constant-cost contours
Figure 5.32. Level curves of function E(x)== 4x. 1
+ x 2
.