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Unformatted text preview: EE236C (Spring 200809) 8. Localization and cuttingplane methods • cuttingplane oracle • finding cuttingplanes • localization algorithms • specific cuttingplane methods • epigraph cuttingplane 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 cuttingplane methods 8–2 Cuttingplane oracle goal: find a point in convex set X ⊆ R n , or determine that X is empty cuttingplane oracle • provides blackbox 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 cuttingplane , or cut , since it eliminates the halfspace { z  a T z > b } from our search for a point in X Localization and cuttingplane 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 cuttingplane methods 8–4 Unconstrained minimization take set of minimizers of f as X cuttingplane oracle (for convex f ): if negationslash = g ∈ ∂f ( x ) , then g T ( z − x ) ≤ defines a (neutral) cut ( a, b ) = ( g, g T x ) at x proof: if g T ( z − x ) > , then z negationslash∈ X f ( z ) ≥ f ( x ) + g T ( z − x ) > f ( x ) Localization and cuttingplane methods 8–5 interpretation g x level curves of f g T ( z − x ) ≥ • 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 cuttingplane methods 8–6 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 ≤ proof: if f ( x ) + g T ( z − x ) − ¯ f > , then z negationslash∈ X f ( z ) ≥ f ( x ) + g T ( z − x ) > ¯ f ≥ f ⋆ Localization and cuttingplane methods 8–7 Feasibility problem X is solution set of convex inequalities f i ( x ) ≤ , i = 1 , . . . , m cuttingplane oracle if x not feasible, find j with f j ( x ) > , and evaluate g j ∈...
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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.
 Spring '08
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