Decomposition and Linearization for 0-1 Quadratic Programming

This paper presents a general decomposition method to compute bounds for constrained 0-1 quadratic programming. The best decomposition is found by using a Lagrangian decomposition of the problem. Moreover, in its simplest version this method is proved to give at least the bound obtained by the LP-relaxation of a non-trivial linearization. To illustrate this point, some computational results are given for the 0-1 quadratic knapsack problem.

[1]  Egon Balas,et al.  Nonlinear 0–1 programming: I. Linearization techniques , 1984, Math. Program..

[2]  Alain Billionnet,et al.  A new upper bound for the 0-1 quadratic knapsack problem , 1999, Eur. J. Oper. Res..

[3]  Franz Rendl,et al.  A recipe for semidefinite relaxation for (0,1)-quadratic programming , 1995, J. Glob. Optim..

[4]  David J. Rader,et al.  Efficient Methods For Solving Quadratic 0–1 Knapsack Problems , 1997 .

[5]  P. Chardaire,et al.  A Decomposition Method for Quadratic Zero-One Programming , 1995 .

[6]  Nelson Maculan,et al.  Lagrangean decomposition for integer nonlinear programming with linear constraints , 1991, Math. Program..

[7]  Alain Faye.  Programmation quadratique en variables bivalentes sous contraintes linéaires. Application au placement de taches dans les systèmes distribués et a la partition de graphes , 1994 .

[8]  Fred W. Glover,et al.  Technical Note - Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program , 1974, Oper. Res..

[9]  Zvi Drezner,et al.  Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming , 1995, Oper. Res..

[10]  Warren P. Adams,et al.  A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems , 1986 .

[11]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[12]  M. Guignard,et al.  Lagrangean decomposition for integer programming: theory and applications , 1987 .

[13]  Egon Balas,et al.  Nonlinear 0–1 programming: II. Dominance relations and algorithms , 1983, Math. Program..

[14]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems , 1994, Discret. Appl. Math..

[15]  Alain Billionnet,et al.  Linear programming for the 0–1 quadratic knapsack problem , 1996 .