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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as max-cut. Other applications include max-min eigenvalue problems and relaxations for the stable set problem.

. This paper describes a software package, called LOQO, which implements a primaldual interior-point method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1. INTRODUCTION LOQO is a software package for solving general (smooth) nonlinear optimization problems. It implements an infeasible-primal-dual path-following method. For linear programming, such methods were first proposed independently by Lust...

this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can

In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasible-primal-dual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasible-primal-dual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2\Theta2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3 \Theta 3-block systems to symmetric 2 \Theta 2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefini...

. We present computational experience with a branch-and-cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables. Such problems arise in financial applications. The algorithm solves the largest real-life problems in a few minutes of run-time. 1 Introduction. We are interested in optimization problems QMIP of the form: min x T Qx + c T x s.t. Ax b (1) jsupp(x)j K (2) 0 x j u j ; all j (3) where x is an n-vector, Q is a symmetric positive-semidefinite matrix, supp(x) = fj : x j ? 0g and K is a positive integer. Problems of this type are of interest in portfolio optimization. Briefly, variables in the problem correspond to commodities to be bought, the objective is a measure of "risk", the constraints (1) prescribe levels of "performance", and constraint (2) specifies that not too many 1 different types of commodities can be chosen. All data is derived from statistical information. A good deal of previous work ha...

Several issues concerning an analysis of large and sparse linear programming problems prior to solving them with an interior point based optimizer are addressed in this paper. Three types of presolve procedures are distinguished. Routines from the first class repeatedly analyze an LP problem formulation: eliminate empty or singleton rows and columns, look for primal and dual forcing or dominated constraints, tighten bounds for variables and shadow prices or just the opposite, relax them to find implied free variables. The second type of analysis aims at reducing a fill-in of the Cholesky factor of the normal equations matrix used to compute orthogonal projections and includes a heuristic for increasing the sparsity of the LP constraint matrix and a technique of splitting dense columns in it. Finally, routines from the third class detect, and remove, different linear dependecies of rows and columns in a constraint matrix. Computational results on problems from the Netlib collection, inc...

... Min-Cost Flow problem, where the mutual capacity constraints are dualized and the resulting Lagrangean Dual is solved with a dual-ascent algorithm belonging to the class of Bundle methods. Although decomposition approaches to block-structured Linear Programs have been reported not to be competitive with general-purpose software, our extensive computational comparison shows that, when carefully implemented, a decomposition algorithm can outperform several other approaches, especially on problems where the number of commodities is “large” with respect to the size of the graph. Our specialized Bundle algorithm is characterized by a new heuristic for the trust region parameter handling, and embeds a specialized Quadratic Program solver that allows the efficient implementation of strategies for reducing the number of active Lagrangean variables. We also exploit the structural properties of the single-commodity Min-Cost Flow subproblems to reduce the overall computational cost. The proposed approach can be easily extended to handle variants of the problem.

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 positive-definite 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 ill-conditioned 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 primal-dual 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.