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06-accpm_notes - Analytic Center Cutting-Plane Method S...

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Analytic Center Cutting-Plane Method S. Boyd, L. Vandenberghe, and J. Skaf April 13, 2008 Contents 1 Analytic center cutting-plane method 2 2 Computing the analytic center 3 3 Pruning constraints 5 4 Lower bound and stopping criterion 5 5 Convergence proof 7 6 Numerical examples 10 6.1 Basic ACCPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6.2 ACCPM with constraint dropping . . . . . . . . . . . . . . . . . . . . . . . . 10 6.3 Epigraph ACCPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1
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In these notes we describe in more detail the analytic center cutting-plane method (AC- CPM) for non-differentiable convex optimization, prove its convergence, and give some nu- merical examples. ACCPM was developed by Goffin and Vial [GV93] and analyzed by Nesterov [Nes95] and Atkinson and Vaidya [AV95]. These notes assume a basic familiarity with convex optimization (see [BV04]), cutting- plane methods (see the EE364b notes Localization and Cutting-Plane Methods ), and subgra- dients (see the EE364b notes Subgradients ). 1 Analytic center cutting-plane method The basic ACCPM algorithm is: Analytic center cutting-plane method (ACCPM) given an initial polyhedron P 0 known to contain X . k := 0. repeat Compute x ( k +1) , the analytic center of P k . Query the cutting-plane oracle at x ( k +1) . If the oracle determines that x ( k +1) X , quit. Else, add the returned cutting-plane inequality to P . P k +1 := P k ∩ { z | a T z b } If P k +1 = , quit. k := k + 1. There are several variations on this basic algorithm. For example, at each step we can add multiple cuts, instead of just one. We can also prune or drop constraints, for example, after computing the analytic center of P k . Later we will see a simple but non-heuristic stopping criterion. We can construct a cutting-plane a T z b at x ( k ) , for the standard convex problem minimize f 0 ( x ) subject to f i ( x ) 0 , i = 1 , . . . , m, as follows. If x ( k ) violates the i th constraint, i.e. , f i ( x ( k ) ) > 0, we can take a = g i , b = g T i x ( k ) f i ( x ( k ) ) , (1) where g i ∂f i ( x ( k ) ). If x ( k ) is feasible, we can take a = g 0 , b = g T 0 x ( k ) f 0 ( x ( k ) ) + f ( k ) best , (2) where g 0 ∂f 0 ( x ( k ) ), and f ( k ) best is the best (lowest) objective value encountered for a feasible iterate. 2
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2 Computing the analytic center Each iteration of ACCPM requires computing the analytic center of a set of linear inequalities (and, possibly, determining whether the set of linear inequalities is feasible), a T i x b i , i = 1 , . . . , m, that define the current localization polyhedron P . In this section we describe some methods that can be used to do this. We note that the inequalities defined by a i and b i , as well as their number m , can change at each iteration of ACCPM, as we add new cutting-planes and possibly prune others. In addition, these inequalities can include some of the original inequalities that define P 0 . To find the analytic center, we must solve the problem minimize Φ( x ) = m i =1 log( b i a T i x ) . (3) This is an unconstrained problem, but the domain of the objective function is the open polyhedron dom Φ = { x | a T i x < b i , i = 1 , . . . , m } , i.e. , the interior of the polyhedron. Part of the challenge of computing the analytic center is that we are not given a point in the domain. One simple approach is to use a phase I optimiza- tion method (see [BV04, § 11.4]) to find a point in
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