Consider solving a linear program in standard form, where the constraint matrix $A$ is $m \times n$, with $n \gg m \gg 1$. Such problems arise, for example, as the result of finely discretizing a semi-infinite program. The cost per iteration of typical primal-dual interior-point methods on such problems is $O(m^2n)$. We propose to reduce … Read more
We present mixed integer linear programming models that are robust in the face of uncertain traffic demands known to lie in a certain polyhedron for the problem of dense wavelength division multiplexing network routing and provisioning at minimal cost. We investigate the solution of the problem in a set of numerical experiments for two models … Read more
Multi-leader-follower games arise when modeling competition between two or more dominant firms and lead in a natural way to equilibrium problems with equilibrium constraints (EPECs). We examine a variety of nonlinear optimization and nonlinear complementarity formulations of EPECs. We distinguish two broad cases: problems where the leaders can cost-differentiate and problems with price-consistent followers. We … Read more
One of the fundamental clustering problems is to assign $n$ points into $k$ clusters based on the minimal sum-of-squares(MSSC), which is known to be NP-hard. In this paper, by using matrix arguments, we first model MSSC as a so-called 0-1 semidefinite programming (SDP). We show that our 0-1 SDP model provides an unified framework for … Read more
Working with an integer bilinear programming formulation of a one-dimensional cutting-stock problem, we develop an ILP-based local-search heuristic. The ILPs holistically integrate the master and subproblem of the usual price driven pattern-generation paradigm, resulting in a unified model that generates new patterns in situ. We work harder to generate new columns, but we are guaranteed … Read more
The Lovasz theta function is a well-known polynomial lower bound on the chromatic number. . Any near optimal solution of its semidefinite programming formulation carries valuable information on how to color the graph. A self-contained presentation of the role of this formulation in obtaining heuristics for the graph coloring problem is presented. CitationSubmitted to Discrete … Read more
The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study … Read more
Several ‘tubechallenges’ have been attempted on the London Underground network, often gaining vast press coverage. The most famous of them consists of trying to travel to all 275 stations of the London Underground network (known as ‘the tube’) in the shortest possible time. In this article we examine how Operational Research can assist potential record-breakers, … Read more
We study a problem faced by a major beverage producer. The company produces and distributes several brands to various customers from its regional distributors. For some of these brands, most customers do not have enough demand to justify full pallet shipments. Therefore, the company decided to design a number of mixed or “rainbow” pallets so … Read more
We study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph. Our work is based on, and refines De Klerk and Pasechnik’s approach to approximating the stability number via copositive programming (SIAM J. Optim. 12 (2002), 875–892). We provide a closed-form expression for the values computed by the … Read more