We present a uni ed framework for studying interior point methods for linear programming. Within this framework, we compare three fundamental methods: (1) pathfollowing, (2) barrier, and (3) primal/dual a ne-scaling (our terminology di ers slightly from the commonly accepted terminology). The step directions for each of these methods are linear combinations of three fundamental directions: (1) ...

This article provides a synopsis of the major developments in interior point methods for mathematical programming in the last twelve 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

This article studies the linear multicommodity network flow problem. This kind of problem arises in a wide variety of contexts. A numerical implementation of the primal-dual interior-point method is designed to solve the problem. In the interior-point method, at each iteration, the corresponding linear problem, expressed as an augmented indefinite system, is solved by using the AINV algorithm c...

This paper studies a joint optimization problem of sub-carrier assignment and power allocation in orthogonal frequency division multiple access (OFDMA) wireless networks. A major challenge in solving the optimization problem is non-convexity caused by the combinatorial nature of sub-carrier assignment problem and/or non-convex objective functions. To address the combinatorial complexity, we for...

This paper addresses the role of centrality in the implementation of interior point methods. We provide theoretical arguments to justify the use of a symmetric neighbourhood, and translate them into computational practice leading to a new insight into the role of recentering in the implementation of interior point methods. Second-order correctors, such as Mehrotra’s predictor–corrector, can occ...

We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discrete-time optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete time linear-quadratic regulator problem with mixed state/control constraints, and show how it can b...

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semide nite programming, and nonconvex and nonlinear problems, have reached varyin...

ABSTRACT. We study and compare preconditioners available for network interior point methods. We derive upper bounds for the condition number of the preconditioned matrices used in the solution of systems of linear equations defining the algorithm search directions. The preconditioners are tested using PDNET, a state-of-the-art interior point code for the minimum cost network flow problem. A com...

We propose the use of controlled perturbations to address the challenging question of optimal active-set prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close t...