This software package is a Matlab implementation of infeasible path-following algorithms for solving standard semidefinite programming (SDP) problems. Mehrotratype predictor-corrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP problem are also implemented. Four types of search directions are available, namely, the AHO, HKM, NT, and GT directions. A few classes of SDP problems are included as well. Numerical results for these classes show that our algorithms are fairly efficient and robust on problems with dimensions of the order of a few hundreds.

under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. ISBN 0-0000-0000-0 The text for this book was formated in Times-Roman and the mathematics was formated in Michael Spivak’s Mathtimes using AMS-L ATEX(which is a macro package for Leslie Lamport’s L ATEX, which itself is a macro package for Donald Knuth’s TEXtext formatting system) and converted from device-independent to postscript format using DVIPS. The fig-ures were produced using SHOWCASE on a Silicon Graphics, Inc. workstation and were incorporated into the text as encapsulated postscript files with the macro package called PSFIG.TEX. To my parents, Howard and Marilyn, my dear wife, Krisadee, and the babes, Marisa and Diana Contents

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...

In this paper, we describe our implementation of a primal-dual infeasible-interior-point algorithm for large-scale linear programming under the MATLAB 1 environment. The resulting software is called LIPSOL -- Linear-programming Interior-Point SOLvers. LIPSOL is designed to take the advantages of MATLAB's sparse-matrix functions and external interface facilities, and of existing Fortran sparse Cholesky codes. Under the MATLAB environment, LIPSOL inherits a high degree of simplicity and versatility in comparison to its counterparts in Fortran or C language. More importantly, our extensive computational results demonstrate that LIPSOL also attains an impressive performance comparable with that of efficient Fortran or C codes in solving large-scale problems. In addition, we discuss in detail a technique for overcoming numerical instability in Cholesky factorization at the end-stage of iterations in interior-point algorithms. Keywords: Linear programming, Primal-Dual infeasible-interior-p...

A simple homogeneous primal-dual feasibility model is proposed for semidefinite programming (SDP) problems. Two infeasible-interior-point algorithms are applied to the homogeneous formulation. The algorithms do not need big M initialization. If the original SDP problem has a solution, then both algorithms find an ffl-approximate solution (i.e., a solution with residual error less than or equal to ffl) in at most O( p n ln(ae ffl 0 =ffl)) steps, where ae is the trace norm of a solution and ffl 0 is the residual error at the (normalized) starting point. A simple way of monitoring possible infeasibility of the original SDP problem is provided such that in at most O( p n ln(aeffl 0 =ffl)) steps either an ffl-approximate solution is obtained, or it is determined that there is no solution with trace norm less than or equal to a given number ae ? 0. Key Words: semidefinite programming, homogeneous interior-point algorithm, polynomial complexity. Abbreviated Title: Homogeneous al...

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...

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...

. We investigate a modified Cholesky algorithm typical of those used in most interiorpoint codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use alternative factorizations; and they perform robustly on most practical problems, yielding good interior-point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill-conditioned. We investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the potential limitations of this approach. Key words. interior-point algorithms and software, Cholesky factorization, matrix p...

We present a generalization of a homogeneous self-dual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "big-M" parameter or two-phase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interior-point and infeasible-starting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and self-dual, infeasible-starting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK-5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...

We describe an algorithm for the monotone linear complementarity problem (LCP) that converges from any positive, not necessarily feasible, starting point and exhibits polynomial complexity if some additional assumptions are made on the starting point. If the problem has a strictly complementary solution, the method converges subquadratically. We show that the algorithm and its convergence properties extend readily to the mixed monotone linear complementarity problem and, hence, to all the usual formulations of the linear programming and convex quadratic programming problems. 1 Introduction The monotone linear complementarityproblem (LCP) is to find a vector pair (x; y) 2 IR n \ThetaIR n such that y = Mx+ q; (x; y) 0; x T y = 0; (1) where q 2 IR n and M is an n \Theta n positive semidefinite (p.s.d.) matrix. The mixed monotone linear complementarity problem (MLCP) is to find a vector triple (x; y; z) 2 IR n \Theta IR n \Theta IR m such that " y 0 # = " M 11 M 12 ...