A Polyhedral Study of the Cardinality Constrained Knapsack Problem
A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of non-negative variables are allowed to be positive. This structure occurs, for example, in areas such as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems.
KeywordsKnapsack Problem Valid Inequality Continuous Formulation Linear Programming Relaxation Cardinality Constraint
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- 3.E.L.M. Beale, “Integer Programming,” in: K. Schittkowski (Ed.), Computational Mathematical Programming, NATO ASI Series, Vol. F15, Springer-Verlag, 1985, pp. 1–24.Google Scholar
- 4.E.L.M. Beale and J.A. Tomlin, “Special Facilities in a General Mathematical Programming System for Nonconvex Problems Using Ordered Sets of Variables,” in: J. Lawrence (Ed.), Proceedings of the fifth Int. Conf. on O.R., Tavistock Publications, 1970, pp. 447–454.Google Scholar
- 5.L.T. Biegler, I.E. Grossmann, and A.W. Westerberg, Systematic Methods of Chemical Process Design, Prentice Hall, 1997.Google Scholar
- 9.I.R. de Farias, Jr., “A Polyhedral Approach to Combinatorial Complementarity Programming Problems,” Ph.D. Thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA (1995).Google Scholar
- 13.I.R. de Farias, Jr., E.L. Johnson, and G.L. Nemhauser, “Facets of the Complementarity Knapsack Polytope,” to appear in Mathematics of Operations Research.Google Scholar
- 15.I.R. de Farias, Jr. and G.L. Nemhauser, “A Polyhedral Study of the Cardinality Constrained Knapsack Problem,” CORE Discussion Paper (2001).Google Scholar
- 21.H. Marchand, A. Martin, R. Weismantel, and L.A. Wolsey, “Cutting Planes in Integer and Mixed-Integer Programming,” CORE Discussion Paper (1999).Google Scholar
- 24.G.L. Nemhauser and L.A. Wolsey, Integer Programming and Combinatorial Optimization, John Wiley and Sons, 1988.Google Scholar
- 29.M.W.P. Savelsbergh, “Functional Description of MINTO, a Mixed INTeger Optimizer (version 3.0),” http://udaloy.isye.gatech.edu/mwps/projects/minto.html.
- 30.P. van Hentenryck, Constraint Satisfaction in Logic Programming, MIT Press, 1989.Google Scholar
- 34.L.A. Wolsey, Integer Programming, John Wiley and Sons, 1998.Google Scholar