Interior point methods for large-scale linear programming

We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned … Read more

A semidefinite programming based polyhedral cut and price algorithm for the maxcut problem

We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the well-known SDP relaxation of maxcut is formulated as a semi-infinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this constitutes the pricing (column generation) phase of the … Read more

Rebalancing an Investment Portfolio in the Presence of Transaction Costs

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. In this paper, we consider an extension of the standard portfolio problem in which transaction costs are incurred to rebalance an investment portfolio. The Markowitz framework of mean-variance efficiency is used with costs modelled as a percentage of the value transacted. … Read more

A semidefinite programming heuristic for quadratic programming problems with complementarity constraints

The presence of complementarity constraints brings a combinatorial flavour to an optimization problem. A quadratic programming problem with complementarity constraints can be relaxed to give a semidefinite programming problem. The solution to this relaxation can be used to generate feasible solutions to the complementarity constraints. A quadratic programming problem is solved for each of these … Read more

A unifying framework for several cutting plane algorithms for semidefinite programming

Cutting plane methods provide the means to solve large scale semidefinite programs (SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching, or the addition of cutting planes. We give a survey of various cutting plane approaches for SDP in this paper. These cutting plane approaches arise from … Read more

Using selective orthonormalization to update the analytic center after the addition of multiple cuts

We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope in Euclidean n-space. This is an important issue that arises at every `stage’ in a cutting plane algorithm. If q cuts are to be added, with q no larger than n, … Read more

Semi-infinite linear programming approaches to semidefinite programming problems

Interior point methods, the traditional methods for the $SDP$, are fairly limited in the sizes of problems they can handle. This paper deals with an $LP$ approach to overcome some of these shortcomings. We begin with a semi-infinite linear programming formulation of the $SDP$ and discuss the issue of its discretization in some detail. We … Read more

Polynomial interior point cutting plane methods

Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the … Read more

A Linear Programming Approach to Semidefinite Programming Problems

Until recently, the study of interior point methods has dominated algorithmic research in semidefinite programming (SDP). From a theoretical point of view, these interior point methods offer everything one can hope for; they apply to all SDP’s, exploit second order information and offer polynomial time complexity. Still for practical applications with many constraints $k$, the … Read more

Realignment in the NFL

The National Football League (NFL) in the United States will expand to 32 teams in 2002 with the addition of a team in Houston. At that point, the league will be realigned into eight divisions, each containing four teams. We describe a branch-and-cut algorithm for minimizing the sum of intradivisional travel distances. We consider first … Read more