5gprojection

5gprojection - EE236C(Spring 2008-09 5 Gradient projection • projected gradient • examples • convergence analysis • dual gradient methods

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Unformatted text preview: EE236C (Spring 2008-09) 5. Gradient projection • projected gradient • examples • convergence analysis • dual gradient methods 5–1 (Sub-)gradient projection minimize f ( x ) subject to x ∈ C f convex with dom f = R n C a closed convex set; we denote by P C (or just P ) the projection on C • gradient projection ( f differentiable) x + = P ( x − t ∇ f ( x )) can also be included in Nesterov’s fast gradient method • subgradient projection ( f nondifferentiable) x + = P ( x − tg ) g ∈ ∂f ( x ) interesting as large-scale algorithms if the projection is inexpensive Gradient projection 5–2 Affine sets hyperplane: C = { x | a T x = b } (with a negationslash = 0 ) P ( x ) = x + b − a T x bardbl a bardbl 2 2 a affine set: C = { x | Ax = b } (with A ∈ R p × n and rank ( A ) = p ) P ( x ) = x + A T ( AA T ) − 1 ( b − Ax ) inexpensive if p ≪ n , or AA T = I , . . . Gradient projection 5–3 Simple polyhedral sets halfspace: C = { x | a T x ≤ b } (with a negationslash = 0 ) P ( x ) = x + b − a T x bardbl a bardbl 2 2 a (if a T x > b ) , P ( x ) = x (otherwise) rectangle: C = { x | l precedesequal x precedesequal u } P ( x ) i = l i x i ≤ l i x i l i ≤ x i ≤ u i u i x i ≥ u i nonnegative orthant: C = R n + P ( x ) = x + ( x + is componentwise max of and x ) Gradient projection 5–4 probability simplex: C = { x | 1 T x = 1 , x followsequal } P ( x ) = ( x − λ 1 ) + where λ is the solution of the equation 1 T ( x − λ 1 ) + = n summationdisplay i =1 max { , x k − λ } = 1 intersection of hyperplane and rectangle: C = { x | a T x = b, l precedesequal x precedesequal u } P ( x ) = P [ l,u ] ( x − λa ) where λ is the solution of a T P [ l,u ] ( x − λa ) = b Gradient projection 5–5 Norm balls Euclidean ball: C = { x | bardbl x bardbl 2 ≤ 1 } P ( x ) = 1 bardbl x bardbl 2 x (if bardbl x bardbl 2 ≥ 1 ) , P ( x ) = x (otherwise) 1-norm ball: C = { x | bardbl x bardbl 1 ≤ 1 } P ( x ) k = x k − λ x k > λ − λ ≤ x k ≤ λ x k + λ x k < − λ λ = 0 if bardbl x bardbl 1 ≤ 1 ; otherwise λ is the solution of the equation n summationdisplay k =1 max {| x k | − λ, } = 1 Gradient projection 5–6 Simple cones second order cone C = { ( x, t...
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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.

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5gprojection - EE236C(Spring 2008-09 5 Gradient projection • projected gradient • examples • convergence analysis • dual gradient methods

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