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8localization

8localization - EE236C(Spring 2008-09 8 Localization and...

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EE236C (Spring 2008-09) 8. Localization and cutting-plane methods cutting-plane oracle finding cutting-planes localization algorithms specific cutting-plane methods epigraph cutting-plane method 8–1 Localization methods based on idea of ‘localizing’ desired point in some set, which becomes smaller at each step like subgradient methods, require one subgradient of objective or constraint functions at each step handle nondifferentiable convex (and quasiconvex) problems typically require more memory and computation per step than subgradient methods but can be much more efficient (in theory and practice) Localization and cutting-plane methods 8–2
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Cutting-plane oracle goal: find a point in convex set X R n , or determine that X is empty cutting-plane oracle provides black-box description of X when oracle is queried at x R n , it either asserts that x X , or returns a separating hyperplane between x and X : a negationslash = 0 , a T z b for z X, a T x b ( a,b ) called a cutting-plane , or cut , since it eliminates the halfspace { z | a T z > b } from our search for a point in X Localization and cutting-plane methods 8–3 Neutral and deep cuts neutral cut: a T x = b ( x is on boundary of halfspace that is cut) deep cut: a T x > b ( x lies in interior of halfspace that is cut) x x X X Localization and cutting-plane methods 8–4
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Unconstrained minimization take set of minimizers of f as X cutting-plane oracle (for convex f ): if 0 negationslash = g ∂f ( x ) , then g T ( z x ) 0 defines a (neutral) cut ( a,b ) = ( g,g T x ) at x proof: if g T ( z x ) > 0 , then z negationslash∈ X f ( z ) f ( x ) + g T ( z x ) > f ( x ) Localization and cutting-plane methods 8–5 interpretation g x level curves of f g T ( z x ) 0 by evaluating g ∂f ( x ) we rule out halfspace in search for optimum get one ‘bit’ of info (on location of x ) by evaluating g Localization and cutting-plane methods 8–6
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Deep cut for unconstrained minimization suppose we know a number ¯ f with f ( x ) > ¯ f f ( e.g. , the smallest value of f found so far in an algorithm) deep cut: if g ∂f ( x ) , then a deep cut is given by g T ( z x ) + f ( x ) ¯ f 0 proof: if f ( x ) + g T ( z x ) ¯ f > 0 , then z negationslash∈ X f ( z ) f ( x ) + g T ( z x ) > ¯ f f Localization and cutting-plane methods 8–7 Feasibility problem X is solution set of convex inequalities f i ( x ) 0 , i = 1 ,...,m cutting-plane oracle if x not feasible, find j with f j ( x ) > 0 , and evaluate g j ∂f j ( x ) ; f j ( x ) + g T j ( z x ) 0 is a deep cut proof: if f j ( x ) + g T j ( z x ) > 0 , then z negationslash∈ X f j ( z ) f j ( x ) + g T j ( z x ) > 0
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