L17_ipm - Lecture 17: Interior-Point Method March 26, 2007...

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Unformatted text preview: Lecture 17: Interior-Point Method March 26, 2007 Lecture 17 Outline • Review of Self-concordance • Overview of Netwon’s Methods for Equality Constrained Minimization • Examples • Interior-Point Method Convex Optimization 1 Lecture 17 Self-Concordance • Overcomes the shortcoming of classical analysis of Newton’s method, where the convergence rate depends on constants m , M , and c : k∇ 2 f ( x )- ∇ 2 f ( y ) k ≤ c k x- y k 2 mI ∇ 2 f ( x ) MI and m , M , and c change with the choice of the norm k · k • Self-concordance introduced by Nesterov and Nemirovski [1994] ∇ 3 f ( x )[ u, u, u ] ≤ M f k u k 3 / 2 ∇ 2 f ( x ) = M f u T ∇ 2 f ( x ) u 3 / 2 Def. A function f : R 7→ R is self-concordant when f is convex and | f 000 ( x ) | ≤ 2 f 00 ( x ) 3 / 2 and for all x ∈ dom f The rate of change in curvature of f is bounded by the curvature Def. A function f : R n 7→ R is self-concordant when its restriction to any line is self-concordant. Note : One can use a constant κ other than 2 in the definition Convex Optimization 2 Lecture 17 Equality Constrained Minimization minimize f ( x ) subject to Ax = b KKT Optimality Conditions imply that x * is optimal if and only if there exists a λ * such that Ax * = b, ∇ f ( x * ) + A T λ * = 0 • Newton’s Method solves KKT conditions " ∇ 2 f ( x k ) A T A #" d k w k # =- " ∇ f ( x k ) h k # where • Feasible point method uses h k = 0 • Infeasible point method uses h k = Ax k- b with w k being dual optimal for the minimization of the quadratic approximation of f at x k Convex Optimization 3 Lecture 17 Equality Constrained Analytic Centering minimize f ( x ) =- ∑ n i =1 ln x i subject to Ax = b Feasible point Newton’s method : g = ∇ f ( x ) , H = ∇ 2 f ( x ) " H A T A #" d w # = "- g # , g = - 1 x 1 . . .- 1 x n , H = diag " 1 x 2 1 , . . . , 1 x 2 n # • The Hessian is positive definite • KKT matrix first row: Hd + A T w =- g ⇒ d =- H- 1 ( g + A T w ) (1) • KKT matrix second row, Ad = 0 , and Eq. (1) ⇒ AH- 1 ( g + A T w ) = 0 • The matrix A has full row rank, thus AH- 1 A T is invertible, hence w =- AH- 1 A T- 1 AH- 1 g, H- 1 = diag h x 2 1 , . . . , x 2 n i • The matrix- AH- 1 A T is known as Schur complement of H (any H ) Convex Optimization 4 Lecture 17 Network Flow Optimization minimize ∑ n l =1 φ l ( x l ) subject to Ax = b • Directed graph with n arcs and p + 1 nodes • Variable x l : flow through arc...
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This note was uploaded on 08/22/2008 for the course GE 498 AN taught by Professor Angelianedich during the Spring '07 term at University of Illinois at Urbana–Champaign.

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L17_ipm - Lecture 17: Interior-Point Method March 26, 2007...

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