lpprox - Math. Program., Ser. A (2008) 112:275301 DOI

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Math. Program., Ser. A (2008) 112:275–301 DOI 10.1007/s10107-006-0017-0 FULL LENGTH PAPER A sparse proximal implementation of the LP dual active set algorithm Timothy A. Davis · William W. Hager Received: 13 August 2003 / Accepted: 21 June 2006 / Published online: 18 August 2006 © Springer-Verlag 2006 Abstract We present an implementation of the LP Dual Active Set Algorithm (LP DASA) based on a quadratic proximal approximation, a strategy for drop- ping inactive equations from the constraints, and recently developed algorithms for updating a sparse Cholesky factorization after a low-rank change. Although our main focus is linear programming, the ±rst and second-order proximal tech- niques that we develop are applicable to general concave–convex Lagrangians and to linear equality and inequality constraints. We use Netlib LP test prob- lems to compare our proximal implementation of LP DASA to Simplex and Barrier algorithms as implemented in CPLEX. Keywords Dual active set algorithm · Linear programming · Simplex method · Barrier method · Interior point method · Equation dropping Mathematics Subject ClassiFcation (2000) 90C05 · 90C06 · 65Y20 This material is based upon work supported by the National Science Foundation under Grant No. 0203270. T. A. Davis Department of Computer and Information Science and Engineering, University of Florida, Gainesville, PO Box 116120, FL 32611-6120, USA e-mail: davis@cise.u².edu URL: http://www.cise.u².edu/ davis W. W. Hager( B ) Department of Mathematics, University of Florida, Gainesville, PO Box 118105, FL 32611-8105, USA e-mail: hager@math.u².edu URL: http://www.math.u².edu/ hager
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276 T. A. Davis, W. W. Hager 1 Introduction We present an implementation of the LP Dual Active Set Algorithm (LP DASA) [23] for solving linear programming problems. Global convergence is established and comparisons with Simplex and Barrier algorithms, as imple- mented in CPLEX, are reported. Since our quadratic proximal approach can be applied to general concave–convex Lagrangians, we develop the theory in an abstract setting, and then apply it to the linear programming problem. Given a function L : 3 × X 7→ R , where 3 R m and X R n are closed and convex, we consider the problem sup λ 3 inf x X L ( λ , x ) .( 1 ) We refer to L in (1) as the “Lagrangian,” while the “dual function,” deFned on 3 , is given by L ( λ ) = inf x X L ( λ , x ) 2 ) Hence, the maximin problem (1) is equivalent to the maximization of the dual function. Let c T denote the transpose of a vector c . The primal linear program (LP) min c T x subject to Ax = b , x 0 ,( 3 ) where A is m by n and all vectors are of compatible size, corresponds to the following choices in the dual formulation (1): X ={ x R n : x 0 } , 3 = R m , and (4) L ( λ , x ) = c T x + λ T ( b Ax ) 5 ) Any LP can be written in the canonical form (3) [37]. If the primal problem (3) has a solution x , then dual problem (1) has a solution; if λ is a solution of (1), then x minimizes L ( λ , · ) over X .
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lpprox - Math. Program., Ser. A (2008) 112:275301 DOI

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