In this paper we study special barrier functions for the convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratic form. We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center … Read more
The revised simplex method is often the method of choice when solving large scale sparse linear programming problems, particularly when a family of closely-related problems is to be solved. Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product. For a significant number of practical problems … Read more
The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n >= 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using … Read more
We study the use of binary variables in reformulating general mixed-integer linear programs. We show that binary reformulations result in problems for which almost all the binary variables replacing a general integer variable need to be explored during branching. We also give computational results on the performance of such reformulations in solving the mixed-integer programs, … Read more
We study how the lift-and-project method introduced by Lov\’az and Schrijver \cite{LS91} applies to the cut polytope. We show that the cut polytope of a graph can be found in $k$ iterations if there exist $k$ edges whose contraction produces a graph with no $K_5$-minor. Therefore, for a graph with $n\ge 4$ nodes, $n-4$ iterations … Read more
AMPL is a language and environment for expressing and manipulating mathematical programming problems, i.e., minimizing or maximizing an algebraic objective function subject to algebraic constraints. AMPL permits separating a model, i.e., a symbolic representation of a class of problems, from the data required to specify a particular problem instance. Once AMPL has a problem instance, … Read more
We analyze the properties that optimization algorithms must possess in order to prevent convergence to non-stationary points for the merit function. We show that demanding the exact satisfaction of constraint linearizations results in difficulties in a wide range of optimization algorithms. Feasibility control is a mechanism that prevents convergence to spurious solutions by ensuring that … Read more
A primal-dual active set method for convex quadratic problems with bound constraints is presented. Based on a guess on the active set, a primal-dual pair $(x,s)$ is computed that satisfies the first order optimality condition and the complementarity condition. If $(x,s)$ is not feasible, a new active set is determined, and the process is iterated. … Read more
We investigate the use of interior point methods for solving quadratic programming problems with a small number of linear constraints where the quadratic term consists of a low-rank update to a positive semi-definite matrix. Several formulations of the support vector machine fit into this category. An interesting feature of these particular problems is the volume … Read more
The term Planning is not used in Operations Research in the sense that is most common in Artificial Intelligence. AI Planning does have many features in common with OR scheduling, sequencing, routing, and assignment problems, however. Current approaches to solving such problems can be broadly classified into four areas: Combinatorial Optimization, Integer Programming, Constraint Programming, … Read more