9accpm

9accpm - EE236C (Spring 2008-09) 9. Analytic center...

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Unformatted text preview: EE236C (Spring 2008-09) 9. Analytic center cutting-plane method analytic center cutting-plane method computing the analytic center pruning constraints lower bound and stopping criterion 91 Analytic center and ACCPM analytic center of a set of inequalities Ax precedesequal b x ac = argmin z m summationdisplay i =1 log( b i a T i z ) analytic center cutting-plane method (ACCPM) localization method that represents P k by set of inequalities A ( k ) , b ( k ) selects analytic center of A ( k ) x precedesequal b ( k ) as query point x ( k +1) Analytic center cutting-plane method 92 ACCPM algorithm outline given an initial polyhedron P = { x | A (0) x precedesequal b (0) } known to contain X repeat for k = 1 , 2 , . . . 1. compute x ( k ) , the analytic center of A ( k- 1) x precedesequal b ( k- 1) 2. query cutting-plane oracle at x ( k ) 3. if x ( k ) X , quit; otherwise, add returned cutting plane a T z b : A ( k ) = bracketleftbigg A ( k- 1) a T bracketrightbigg , b ( k ) = bracketleftbigg b ( k- 1) b bracketrightbigg if P k = { x | A ( k ) x precedesequal b ( k ) } = , quit Analytic center cutting-plane method 93 Constructing cutting-planes minimize f ( x ) subject to f i ( x ) , i = 1 , . . . , m f , . . . , f m : R n R convex cutting-plane for optimal set X if x ( k ) is not feasible, say f j ( x ( k ) ) > , we have (deep) feasibility cut f j ( x ( k ) ) + g T j ( z x ( k ) ) , g j f j ( x ( k ) ) if x ( k ) is feasible, we have (deep) objective cut g T ( z x ( k ) ) + f ( x ( k ) ) f ( k ) best , g f ( x ( k ) ) where f ( k ) best = min { f ( x ( i ) ) | i k, x ( i ) feasible } Analytic center cutting-plane method 94 Computing the analytic center minimize ( x ) = m summationdisplay i =1 log( b i a T i x ) dom = { x | a T i x < b i , i = 1 , . . . , m } challenge : we are not given a point in dom some options use phase I to find x dom , followed by standard Newton method standard Newton method applied to dual problem infeasible start Newton method (EE236B lecture 11, BV 10.3) Analytic center cutting-plane method 95 Dual Newton method dual analytic centering problem...
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9accpm - EE236C (Spring 2008-09) 9. Analytic center...

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