Disjunctive Cut Generation for Mixed 0-1 Programs: Duality and Lifting

We analyse a class of mathematical programs that enable the generation of disjunctive cutting planes for mixed 0--1 programming. Through duality theory we provide a natural geometric interpretation of these problems. A simplification step allows us to solve the mathematical programs in a lower dimensional space. The optimal solution is then lifted to the original space of variables. 1 Introduction There has recently been a renewed interest in mixed 0--1 linear programming models with no underlying combinatorial structure. The main goal of this line of research in integer programming is to develop solution techniques, based on cutting plane algorithms, that do not rely on a problemspecific analysis of the polyhedral structure of the integer program. Disjunctive cutting planes rely solely on the fact that some of the variables are integer constrained and have proven to be quite effective when used within a branch-and-cut algorithm. These cuts are selected among the valid inequalities of ..

Convex Programming for Disjunctive Convex Optimization

Given a finite number of closed convex sets whose algebraic representation is known, we study the problem of optimizing a convex function over the closure of the convex hull of the union of these sets. We derive an algebraic characterization of the feasible region in a higher-dimensional space and propose a solution procedure akin to the interior-point approach for convex programming. Research partly supported by NSF HPCC Grant DMS 95-27124. Author's address: 417 Uris Hall, Graduate School of Business, Columbia University, New York NY 10027. E-mail [email protected]. Also affiliated with Computational Optimization Research Center (CORC), Columbia University. y Supported by Subprograma Ciencia e Tecnologia do 2 o Quadro comunit'ario de Apoio grant Praxis XXI/BD/2831/94. Author's address: 804 Uris Hall, Graduate School of Business, Columbia University, New York NY 10027. E-mail [email protected]. 1 Introduction The literature in optimality condition..

Cutting Planes for Integer Programs with General Integer Variables

We investigate the use of cutting planes for integer programs with general integer variables. We show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0-1 variables. We also explore the use of Gomory's mixed integer cuts. We address both theoretical and computational issues and show how to embed these cutting planes in a branch-and-bound framework. We compare results obtained by using our cut generation routines in two existing systems with a commercially available branch-and-bound code on a range of test problems arising from practical applications

OCTANE: A New Heuristic for Pure 0-1 Programs

We propose a new heuristic for pure 0--1 programs, which finds feasible integer points by enumerating extended facets of the octahedron, the outer polar of the unit hypercube. We give efficient algorithms to carry out the enumeration, and explain how our heuristic can be embedded in a branch-and-cut framework. Finally, we present computational results on a set of pure 0--1 programs from MIPLIB. 1 Introduction There is clearly a renewed interest in the research community in computational integer programming. The recent success of branch-and-cut as a solution framework for general integer programs, has revived an area that for a long time had fallen out of favor. Most of this success, however, has been obtained in the area of finding exact solutions for this class of problems, while very little effort has focused on the approximate solution of general 0--1 programs. In fact, the literature in heuristics for general integer programs is quite limited [6, 11, 14]. The only heuristic which..

OCTANE: A New Heuristic for Pure 0-1 Programs

We propose a new heuristic for pure 0--1 programs, which finds feasible integer points by enumerating extended facets of the octahedron, the outer polar of the unit hypercube. We give efficient algorithms to carry out the enumeration, and explain how our heuristic can be embedded in a branch-and-cut framework. Finally, we present computational results on a set of pure 0-1 programs from MIPLIB