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A New Bound for the Quadratic Assignment Problem Based on Convex Quadratic Programming
 MATHEMATICAL PROGRAMMING
, 1999
"... We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be comp ..."
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Cited by 33 (4 self)
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We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoff between bound quality and computational effort.
Solving reduced KKT systems in barrier methods for linear and quadratic programming
, 1991
"... In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite. For linear programs, these KKT systems are usually reduced to smaller positivedefinite systems AH −1 A T q = s, where H is a large principal submatrix of ..."
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Cited by 22 (7 self)
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In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite. For linear programs, these KKT systems are usually reduced to smaller positivedefinite systems AH −1 A T q = s, where H is a large principal submatrix of K. These systems can be solved more efficiently, but AH −1 A T is typically more illconditioned than K. In order to improve the numerical properties of barrier implementations, we discuss the use of “reduced KKT systems”, whose dimension and condition lie somewhere in between those of K and AH −1 A T. The approach applies to linear programs and to positive semidefinite quadratic programs whose Hessian H is at least partially diagonal. We have implemented reduced KKT systems in a primaldual algorithm for linear programming, based on the sparse indefinite solver MA27 from the Harwell Subroutine Library. Some features of the algorithm are presented, along with results on the netlib LP test set.
Why a Pure Primal Newton Barrier Step May Be Infeasible
 SIAM Journal on Optimization
, 1993
"... Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when ..."
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Cited by 21 (3 self)
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Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when the barrier parameter is reduced at a reasonable rate. We present local analysis showing why a pure Newton step in a longstep barrier method for nonlinearly constrained optimization may be seriously infeasible, even when taken from an apparently favorable point. The features described are illustrated numerically and connected to known theoretical results for convex problems satisfying selfconcordancy assumptions. We also indicate the contrasting nature of an approximate step to the desired minimizer of the barrier function. 1. Introduction 1.1. Background Interior methods, most commonly based on barrier functions, have been applied with great practical success in recent years to many con...
The Optimal Set and Optimal Partition Approach to Linear and Quadratic Programming
 in Advances in Sensitivity Analysis and Parametric Programming
, 1996
"... In this chapter we describe the optimal set approach for sensitivity analysis for LP. We show that optimal partitions and optimal sets remain constant between two consecutive transitionpoints of the optimal value function. The advantage of using this approach instead of the classical approach (usin ..."
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Cited by 6 (3 self)
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In this chapter we describe the optimal set approach for sensitivity analysis for LP. We show that optimal partitions and optimal sets remain constant between two consecutive transitionpoints of the optimal value function. The advantage of using this approach instead of the classical approach (using optimal bases) is shown. Moreover, we present an algorithm to compute the partitions, optimal sets and the optimal value function. This is a new algorithm and uses primal and dual optimal solutions. We also extend some of the results to parametric quadratic programming, and discuss differences and resemblances with the linear programming case.
Basis and Tripartition Identification for Quadratic Programming and Linear Complementarity Problems  From an interior solution to an optimal basis and viceversa
, 1996
"... Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal complementary solutions. Maximal complementary solutions can be characterized by optimal (tri)partitions. On the other hand, the solutions provided by simplexb ..."
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Cited by 3 (2 self)
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Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal complementary solutions. Maximal complementary solutions can be characterized by optimal (tri)partitions. On the other hand, the solutions provided by simplexbased pivot algorithms are given in terms of complementary bases. A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A tripartition identification algorithm is an algorithm which generates a maximal complementary solution (and its corresponding tripartition), starting from any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal...
Computing Maximum Likelihood Estimators of Convex Density Functions
, 1995
"... We consider the problem of estimating a density function that is known in advance to be convex. The maximum likelihood estimator is then the solution of a linearly constrained convex minimization problem. This problem turns out to be numerically difficult. We show that interior point algorithms p ..."
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Cited by 2 (0 self)
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We consider the problem of estimating a density function that is known in advance to be convex. The maximum likelihood estimator is then the solution of a linearly constrained convex minimization problem. This problem turns out to be numerically difficult. We show that interior point algorithms perform well on this class of optimization problems, though for large samples, numerical difficulties are still encountered. To eliminate those difficulties, we propose a clustering scheme that is reasonable from a statistical point of view. We display results for problems with up to 40000 observations. We also give a typical picture of the estimated density: a piece wise linear function, with very few pieces only. Key words: interiorpoint method, convex estimation, maximum likelihood estimation, logarithmicbarrier method, primaldual method. iv 1 Introduction Finding a good statistical estimator can often be formulated as an unconstrained optimization problem whose objective func...
An InteriorPoint Method for General LargeScale Quadratic Programming Problems
 Annals of Operations Research
, 1996
"... In this paper we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems, efficiently solves a wide class of large scale problems and forms the basis for a sequential qua ..."
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Cited by 1 (0 self)
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In this paper we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems, efficiently solves a wide class of large scale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a 3dimensional costimprovement subproblem, which is solved at every iteration. We have developed an approximate recentering procedure and a novel, adaptive bigM Phase I procedure that are essential to the success. We describe the basic method along with the recentering and bigM Phase I procedures. Details of the implementation and computational results are also presented. Keywords: bigM Phase I procedure, convex quadratic programming, interior point methods, linear programming, method of centers, multidirectional search direction, nonconvex quadratic programming, recentering. # Cont...