TrustRegion

TrustRegion - Stanford University Trust-Region Methods D....

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Stanford University PE284 Trust-Region Methods D. Echeverría Ciaurri Optimization PE284 Stanford University
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Stanford University Trust-Region Methods Nov 3 2008 PE284 2 Outline Basic Idea The Cauchy Point Algorithms Convergence Constrained Problems Summary
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Stanford University Trust-Region Methods Nov 3 2008 PE284 3 Trust-Region Methods Optimization with safeguards Dual to line search Related to Levenberg ‘Trust Region’ coined by Dennis Applied in great number of fields Included in MATLAB’s fmincon
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Stanford University Trust-Region Methods Nov 3 2008 PE284 4 Main Idea (figures from http://www.applied-mathematics.net )
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Stanford University Trust-Region Methods Nov 3 2008 PE284 5 Main Idea (figures from http://www.applied-mathematics.net )
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Stanford University Trust-Region Methods Nov 3 2008 PE284 6 Main Idea (figures from http://www.applied-mathematics.net )
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Stanford University Trust-Region Methods Nov 3 2008 PE284 7 Main Idea (figures from http://www.applied-mathematics.net )
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Stanford University Trust-Region Methods Nov 3 2008 PE284 8 Main Idea (figures from http://www.applied-mathematics.net )
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Stanford University Trust-Region Methods Nov 3 2008 PE284 9 Main Idea (figures from http://www.applied-mathematics.net )
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Stanford University Trust-Region Methods Nov 3 2008 PE284 10 Main Idea (figures from http://www.applied-mathematics.net )
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Stanford University Trust-Region Methods Nov 3 2008 PE284 11 Trust-Region Step Unconstrained minimization of f(x) with B k symmetric uniformly bounded Solve approximately where k >0 is the trust-region radius ( 29 k 2 k T 2 1 T k k k R p || p || s.t. p B p p f f p m min n + + = ( 29 ( 29 ( 29 k 2 k k k k k k x f B , x f f , x f f , x = =
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Stanford University Trust-Region Methods Nov 3 2008 PE284 12 Trust-Region Step Step p k acceptance + TR control by ( 29 ( 29 ( 29 ( 29 k k k k k k k p m 0 m p x f x f - + - = ρ actual predicted
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Stanford University Trust-Region Methods Nov 3 2008 PE284 13 Trust-Region Step Step p k acceptance + TR control by If ρ k ~ 1 or larger: accept + expand TR If ρ k medium: accept If ρ k small: reject + shrink TR ( 29 ( 29 ( 29 ( 29 k k k k k k k p m 0 m p x f x f - + - = ρ actual predicted
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Stanford University Trust-Region Methods Nov 3 2008 PE284 14 Basic Algorithm Given MAX >0, 0 and 0 ≤ η < η 1 < η 2 and γ 1 < 1 < γ 2 : for i = 0,1,2,… Obtain p k by approx. solving min m k (p) in TR Evaluate ρ k if ρ k < η 1 k+1 = γ 1 || p k || else if ρ k > η 2 and k+1 = || p k || k+1 = min( γ 2 k , , MAX ) else k+1 = k if ρ k > η x k+1 = x k + p k else x k+1 = x k end(for);
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Stanford University Trust-Region Methods Nov 3 2008 PE284 15 Outline Basic Idea The Cauchy Point Algorithms Convergence Constrained Problems Summary
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Stanford University Trust-Region Methods Nov 3 2008 PE284 16 The Cauchy Point Crucial for
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This note was uploaded on 01/24/2011 for the course ERE 284 taught by Professor . during the Spring '10 term at Stanford.

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TrustRegion - Stanford University Trust-Region Methods D....

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