10ellipsoid

10ellipsoid - EE236C (Spring 2008-09) 10. Ellipsoid method...

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Unformatted text preview: EE236C (Spring 2008-09) 10. Ellipsoid method ellipsoid method convergence proof inequality constraints 101 Ellipsoid method history developed by Shor, Nemirovski, Yudin in 1970s used in 1979 by Khachian to show polynomial solvability of LPs properties each step requires cutting-plane or subgradient evaluation modest storage ( O ( n 2 ) ) modest computation per step ( O ( n 2 ) ), via analytical formula extremely simple to implement efficient in theory slow but steady in practice; rarely used Ellipsoid method 102 Motivation drawbacks of cutting-plane methods serious computation needed to find next query point typically, O ( n 2 m ) for analytic centering in ACCPM, with m inequalities localization polyhedron grows in complexity as algorithm progresses (with pruning, can keep m proportional to n , e.g. , m = 4 n ) ellipsoid method addresses both issues, but retains theoretical efficiency Ellipsoid method 103 Ellipsoid algorithm for minimizing convex function idea: localize x in an ellipsoid instead of a polyhedron given an initial ellipsoid E known to contain X repeat for k = 1 , 2 , . . . 1. query oracle to get a neutral cut a T z b at x ( k )) , the center of E k 1 2. set E k := minimum volume ellipsoid covering E k 1 { z | a T z b } E k 1 x ( k ) a E k Ellipsoid method 104 differences with cutting-plane methods localization set doesnt grow more complicated generating query point is trivial but, we add unnecessary points in step 2 interpretation reduces to bisection for n = 1 can be viewed as an implementable version of the center-of-gravity cutting-plane method Ellipsoid method 105 Example a112 x (0) a112 x (1) a112 x (2) a112 x (3) a112 x (4) a112 x (5) Ellipsoid method 106 Updating the ellipsoid...
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This note was uploaded on 01/25/2010 for the course EE 236 taught by Professor Staff during the Spring '08 term at UCLA.

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10ellipsoid - EE236C (Spring 2008-09) 10. Ellipsoid method...

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