Bilevel programming and the separation problem

In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is that of generating a valid inequality violated by a given real vector, usually arising as the solution to a relaxation of the original problem. We show that the problem of generating a maximally violated valid inequality often has a natural interpretation as a bilevel program. In some cases, this bilevel program can be easily reformulated as a simple single-level mathematical program, yielding a standard mathematical programming formulation for the separation problem. In other cases, no such polynomial-size single-level reformulation exists unless the polynomial hierarchy collapses to its first level (an event considered extremely unlikely in computational complexity theory). We illustrate our insights by considering the separation problem for two well-known classes of valid inequalities. Keywords: Bilevel programming; Cutting planes; Separation; Computational complexit

A Grid-Aware Branch, Cut and Price Implementation

This paper presents a grid-enabled system for solving large-scale optimization problems. The system has been developed using Globus and MPICH-G2 grid technologies, and consists of two BCP solvers and of an interface portal. After a brief introduction to Branch, Cut and Price optimization algorithms, the system architecture, the solvers and the portal user interface are described. Finally, some of the tests performed and the obtained results are illustrated

New fast heuristics for the Two-Echelon Vehicle Routing Problem,

In this paper we address the Two-Echelon Vehicle Routing Problem (2E-VRP), an extension of the classical Capacitated VRP, where the delivery from a single depot to the customers is managed by routing and consolidating the freight through intermediate depots called satellites. We present a family of Multi-Start heuristics based on separating the depot-to-satellite transfer and the satellite-to-customer delivery by iteratively solving the two resulting routing subproblems, while adjusting the satellite workloads that link them. The common scheme on which all the heuristics are based consists in, after having found an initial solution, applying a local search phase, followed by a diversification; if the new obtained solutions are feasible, then local search is applied again, otherwise a feasibility search procedure is applied, and if it successful, the local search is applied on the newfound solution. Different diversification strategies and feasibility search rules are proposed. We present computational results on a wide set of instances up to 50 customers and 5 satellites and compare them with results from the literature, showing how the new methods outperform previous existent methods, both in efficiency and accurac

Minimal load constrained vehicle routing problems

In this paper, the Capacitated Vehicle Routing Problem is extended to the case where each vehicle is restricted to an additional minimal starting or returning load constraint. We refer to this extension as the Minimal Load Constrained Vehicle Routing Problem. We present integer programming formulations for single and multidepot cases. An illustrative example is also provided to show how a decision maker can use the proposed formulation as an aid in distribution plannin