E. Polak's Computational Methods in Optimization: A Unified Approach PDF

By E. Polak

ISBN-10: 008096091X

ISBN-13: 9780080960913

ISBN-10: 0125593503

ISBN-13: 9780125593502

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Extra resources for Computational Methods in Optimization: A Unified Approach

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Step 0. Select a step size procedure 9’from (33) and (36), and fix its parameters. * Step 1. Select a z,, E R” such that the set defined by (2) is bounded. Step 2. Set i = 0. * First check whether hi = hi-l is admissible. It is possible that the step size hi may become a constant for all i larger than some k. 53). 38 2 UNCONSTRAINED MINIMIZATION Step 3. Compute Vfo(z,). Step 4. If Vf0(z,)= 0, stop; else, go to step 5. Step 5 . Compute a A, by means of 8,with h(z,) = -Vf@(z,>. m Step 6 . Set z , , ~ = z, - A,Vfo(z,), set i = i 1 , and go to step 3.

Select a z, in R". Step 1. Set i = 0. Step 2. Compute Vfo(zi). Step 3. If Vfo(zi)= 0, stop; else, compute zi+l according to (38), m set i = i 1, and go to step 2. + 40 Exercise. Use the continuity of the Newton-Raphson iteration function (38) to show that if the sequence {zi} constructed by (39) converges to a point z, then VfO(z) = 0. 2. 1 GRADIENT AND QUASI-NEWTON METHODS IN [w" 39 of the following modification, due to Goldstein [G3], which results in the same rate of convergence as the original method (39).

C(Zi+l) - 4 Z i ) l < 8, which shows again that the sequence { c ( z ~ )is} not ~ ~ ~Cauchy and hence cannot converge. However, this again contradicts assumption (i) of (27), } ~ ~converge. ~ We therefore since because of (i) in (27), the sequence { c ( z ~ )must conclude that if 5 is an accumulation point of {zi}, it must be desirable. ” because of the cost of evaluating c(y). ). We shall now describe a model which was constructed by the author in collaboration with Robert Klessig, a graduate student.

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Computational Methods in Optimization: A Unified Approach by E. Polak

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