Applied nonlinear programming by Sanjay Sharma

By Sanjay Sharma

Sharma S. utilized Nonlinear Programming (New Age foreign (P) Ltd, 2006)(ISBN 812241771X)

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Mi = 0, k = 1, 2, ... 6) i =1 Let x1*, x2*, ... 5). F. , N f* = ∑M * i = M1* + M*2 + ... + M*N i =1 Dividing by f* on both sides, M1* M*2 M*N f* + + ... + = =1 f* f* f* f* or w1 + w2 + ... 7) i=1 where wi = M*i f* Mi* = ωi . 6) can be written as, N ∑a * ki . Mi i =1 = 0, substituting the optimum values and GEOMETRIC PROGRAMMING 51 multiplying with xk* on both sides. 9) in above equation, N f* ∑a ki . wi =0 i=1 N ∑a ki . 10) are used to solve the problem. A unique solution is obtained for wi, i = 1, 2,....

Fig. 4(c) – xL and xR on one side of the optimum and f(xL) > f(xR). Following statement is applicable in all the three situations: “Optimum should lie in the range [x1, xR] if f(xL) = f(xR)”. Observe this and the previous statement. Out of xL and xR, only one is changing for the range in which an optimum ONE VARIABLE OPTIMIZATION 23 may lie. This discussion is useful for implementing an algorithm for method of golden section. (a) f(xL) = f(xR) (b) xL and xR on either side of the optimum and f(xL) > f(xR) (c) xL and xR on one side of the optimum and f(xL) > f(xR) Fig.

4(b) – xL and xR on either side of the optimum and f(xL) > f(xR). Fig. 4(c) – xL and xR on one side of the optimum and f(xL) > f(xR). Following statement is applicable in all the three situations: “Optimum should lie in the range [x1, xR] if f(xL) = f(xR)”. Observe this and the previous statement. Out of xL and xR, only one is changing for the range in which an optimum ONE VARIABLE OPTIMIZATION 23 may lie. This discussion is useful for implementing an algorithm for method of golden section. (a) f(xL) = f(xR) (b) xL and xR on either side of the optimum and f(xL) > f(xR) (c) xL and xR on one side of the optimum and f(xL) > f(xR) Fig.

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