Lecture14

Lecture14 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 14 Dr. Ted Ralphs IE417 Lecture 14 1 Reading for This Lecture Sections 8.1-8.5 IE417 Lecture 14 2 One-dimensional Line Search One-dimensional line search is the fundamental subproblem for many non-linear algorithms. Given a function f , a current location x , and a direction d , we want to solve the following problem min f ( x + d ) s.t. a b Recall the typical iterative algorithm discussed in Chapter 7. IE417 Lecture 14 3 Line Search Methods Exact Methods Solve the line search problem analytically. Take the derivative with respect to and set it to zero. Iterative Methods Methods using function evaluations. Methods using derivatives. Generally guaranteed to converge for pseudoconvex functions. IE417 Lecture 14 4 The Interval of Uncertainty The interval of uncertainty is the interval within which the optimal solution has to lie. Most derivative-free line search methods are based on iteratively reducing the interval of uncertainty. Theorem 1. Let : R R be strictly quasiconvex over the interval [ a,b ] . Let , [ a,b ] be such that < . If ( ) > ( ) , then ( z ) ( ) for all z [ a, ) ....
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Lecture14 - Advanced Mathematical Programming IE417 Lecture...

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