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 9–1 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 9–2 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 9–3 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 9–4 Computing the analytic center minimize φ ( x ) = − m summationdisplay i =1 log( b i − a T i x ) dom φ = { x | a T i x < b i , i = 1 , . . . , m } challenge : we are not given a point in dom φ some options • use phase I to find x ∈ dom φ , 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 9–5 Dual Newton method dual analytic centering problem...
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9accpm - EE236C(Spring 2008-09 9 Analytic center...

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