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Unformatted text preview: Lecture 17: InteriorPoint Method March 26, 2007 Lecture 17 Outline • Review of Selfconcordance • Overview of Netwon’s Methods for Equality Constrained Minimization • Examples • InteriorPoint Method Convex Optimization 1 Lecture 17 SelfConcordance • 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 • Selfconcordance 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 selfconcordant 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 selfconcordant when its restriction to any line is selfconcordant. 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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 Spring '07
 AngeliaNedich
 Linear Programming, Optimization, Convex Optimization, Narendra Karmarkar

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