Strong valid inequalities for orthogonal disjunctions and bilinear covering sets
- 287 Downloads
In this paper, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when this convex hull is completely determined by orthogonal restrictions of the original set. Although the tools used in this construction include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and finding the convex hull of certain mixed/pure-integer bilinear sets. We then extend the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the non-negative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant. Although we illustrate our results primarily on bilinear covering sets, they also apply to more general polynomial covering sets for which they yield new tight relaxations.
Mathematics Subject Classification (2000)46N10 90C11 90C26
Unable to display preview. Download preview PDF.
- 4.Balas E.: Disjunctive programming: cutting planes from logical conditions. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds) Nonlinear Programming, pp. 279–312. Academic Press, New York (1975)Google Scholar
- 7.Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44, original manuscript was published as a technical report in 1974 (1998)Google Scholar
- 12.Belotti, P., Lee, J., Liberti, L., Margot, F., Waechter, A.: Branching and bounds tightening techniques for non-convex MINLP. http://www.optimization-online.org/DB_HTML/2008/08/2059.html (2008)
- 13.Bliek, C., Jermann, C., Neumaier, A. (eds.): Global Optimization and Constraint Satisfaction, 5th Annual Workshop on Global Constraint Optimization and Constraint Satisfaction, COCOS, Springer (2002)Google Scholar
- 21.LINDO Systems Inc: LINGO 11.0 optimization modeling software for linear, nonlinear, and integer programming. http://www.lindo.com (2008)
- 25.Sahinidis, N.V., Tawarmalani, M.: BARON. The Optimization Firm, LLC, Urbana-Champaign, IL. http://www.gams.com/dd/docs/solvers/baron.pdf (2005)
- 33.Tawarmalani, M., Richard, J.P.P., Chung, K.: Strong Valid Inequalities for Orthogonal Disjunctions and Polynomial Covering Sets, Technical Report, Krannert School of Management, Purdue University (2008)Google Scholar
- 34.Ziegler G.M.: Lectures on Polytopes. Springer, New York (1998)Google Scholar