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 + λd ) s.t. a ≤ λ ≤ 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 λ 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 Θ : R → R be strictly quasiconvex over the interval [ a,b ] . Let λ,μ ∈ [ a,b ] be such that λ < μ . – If Θ( λ ) > Θ( μ ) , then Θ( z ) ≥ Θ( μ ) for all z ∈ [ a,λ ) ....
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Lecture14 - Advanced Mathematical Programming IE417 Lecture...

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