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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 10–1 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 10–2 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 10–3 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 10–4 differences with cutting-plane methods • localization set doesn’t 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 10–5 Example a112 x (0) a112 x (1) a112 x (2) a112 x (3) a112 x (4) a112 x (5) Ellipsoid method 10–6 Updating the ellipsoid...
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10ellipsoid - EE236C(Spring 2008-09 10 Ellipsoid method •...

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