Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.

We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng

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This article provides a synopsis of the major developments in interior point methods for mathematical programming in the last thirteen years, and discusses current and future research directions in interior point methods, with a brief selective guide to the research literature. AMS Subject Classification: 90C, 90C05, 90C60 Keywords: Linear Programming, Newton's Method, Interior Point Methods, Barrier Method, Semidefinite Programming, Self-Concordance, Convex Programming, Condition Numbers 1 An earlier version of this article has previously appeared in OPTIMA -- Mathematical Programming Society Newsletter No. 51, 1996 2 M.I.T. Sloan School of Management, Building E40-149A, Cambridge, MA 02139, USA. email: rfreund@mit.edu 3 The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106 JAPAN. e-mail: mizuno@ism.ac.jp INTERIOR POINT METHODS 1 1 Introduction and Synopsis The purpose of this article is twofold: to provide a synopsis of the major developments in ...

A simple column generation scheme that employs an interior point method to solve underlying restricted master problems is presented. In contrast with the classical column generation approach where restricted master problems are solved exactly, the method presented in this paper consists in solving it to a predetermined optimality tolerance (loose at the beginning and appropriately tightened when the optimum is approached). An infeasible primal-dual interior point method which employs the notion of ¯-center to control the distance to optimality is used to solve the restricted master problem. Similarly to the analytic center cutting plane method, the present approach takes full advantage of the use of central prices. Furthermore, it offers more freedom in the choice of optimization strategy as it adaptively adjusts the required optimality tolerance in the master to the observed rate of convergence of the column generation process. The proposed method has been implemented and used to solv...

A new primal-dual algorithm is proposed for the minimization of non-convex objective functions subject to general inequality and linear equality constraints. The method uses a primal-dual trustregion model to ensure descent on a suitable merit function. Convergence is proved to second-order critical points from arbitrary starting points. Preliminary numerical results are presented.

Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primal-dual interior-point methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NP-hard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to non-convex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include non-convex potential reduction methods, interior point cutting plane methods, primal-dual IPMs and first-order algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.

We show that the effects of finite-precision arithmetic in forming and solving the linear system that arises at each iteration of primal-dual interior-point algorithms for nonlinear programming are benign. When we replace the standard assumption that the active constraint gradients are independentby the weaker Mangasarian-Fromovitz constraint qualifiation, rapid convergence usually is attainable, even when cancellation and roundoff errors occur during the calculations. In deriving our main results, we proveakey technical result about the size of the exact primal-dual step. This result can be used to modify existing analysis of primal-dual interior-point methods for convex programming, making it possible to extend the superlinear local convergence results to the nonconvex case.

During the last decade interior-point methods have become an efficient alternative to the simplex algorithm for solution of large-scale linear programming (LP) problems. However, in many practical applications of LP, interior-point methods have the drawback that they do not generate an optimal basic and nonbasic partition of the variables. This partition is required in the traditional sensitivity analysis and is highly useful when a sequence of related LP problems are solved. Therefore, in this paper we discuss how an optimal basic solution can be generated from the interior-point solution. The emphasis of the paper is on how problem structure can be exploited to reduce the computational cost associated with the basis identification. Computational results are presented which indicate that it is highly advantageous to exploit problem structure. Key words: Linear programming, interior-point methods, basis identification. 1 Introduction Since the late forties the simplex algorithm has be...

This paper addresses the issues involved with an interior point-based decomposition applied to the solution of linear programs with a block-angular structure. Unlike classical decomposition schemes that use the simplex method to solve subproblems, the approach presented in this paper employs a primal-dual infeasible interior point method. The abovementioned algorithm offers a perfect measure of the distance to optimality, which is exploited to terminate the algorithm earlier (with a rather loose optimality tolerance) and to generate ffl-subgradients. In the decomposition scheme, subproblems are sequentially solved for varying objective functions. It is essential to be able to exploit the optimal solution of the previous problem when solving a subsequent one (with a modified objective). A warm start routine is described that deals with this problem. The proposed approach has been implemented within the context of two optimization codes freely available for research use: the Analytic Ce...