Abstract
In this paper, we study \(0\mathord {-}1\) mixed-integer bilinear covering sets. We derive several families of facet-defining inequalities via sequence-independent lifting techniques. We then show that these sets have a polyhedral structure that is similar to that of a certain fixed-charge single-node flow set. As a result, we also obtain new facet-defining inequalities for the single-node flow set that generalize well-known lifted flow cover inequalities from the integer programming literature.
References
Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)
Atamtürk, A.: Flow pack facets of the single node fixed-charge flow polytope. Oper. Res. Lett. 29, 107–114 (2001)
Atamtürk, A.: On the facets of the mixed-integer knapsack polyhedron. Math. Program. 98, 145–175 (2003)
Balas, E.: Facets of the knapsack polytope. Math. Program. 8, 146–164 (1975)
Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89, 3–44 (original manuscript was published as a technical report in 1974) (1998)
Chaovalitwongse, W., Pardalos, P.M., Prokopyev, O.A.: A new linearization technique for multi-quadratic 0–1 programming problems. Oper. Res. Lett. 32, 517–522 (2004)
Christof, T., Löbel, A.: PORTA: POlyhedron Representation Transformation Algorithm. Available at http://www.zib.de/Optimization/Software/Porta/ (1997)
Chung, K.: Strong valid inequalities for mixed-integer nonlinear programs via disjunctive programming and lifting. PhD thesis, University of Florida, Gainesville, FL (2010)
Chung, K., Richard, J.-P.P., Tawarmalani, M.: Lifted Inequalities for 0–1 Mixed-Integer Bilinear Covering Sets. Technical Report 1272, Krannert School of Management, Purdue University (2011)
Falk, J.E., Soland, R.M.: An algorithm for separable nonconvex programming problems. Manag. Sci. 15, 550–569 (1969)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted flow cover inequalities for mixed \(0\mathord {-}1\) integer programs. Math. Program. 85, 439–467 (1999)
Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Sequence independent lifting in mixed integer programming. J. Comb. Optim. 4, 109–129 (2000)
Hammer, P.L., Johnson, E.L., Peled, U.N.: Facets of regular \(0\mathord {-}1\) polytopes. Math. Program. 8, 179–206 (1975)
Hardy, G., Littlewood, J., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1988)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)
LINDO Systems Inc.: LINGO 11.0 Optimization Modeling Software for Linear, Nonlinear, and Integer Programming. Available at http://www.lindo.com (2008)
Louveaux, Q., Wolsey, L.A.: Lifting, superadditivity, mixed integer rounding and single node flow sets revisited. Ann. Oper. Res. 153, 47–77 (2007)
Marchand, H., Wolsey, L.A.: The \(0\mathord {-}1\) knapsack problem with a single continuous variable. Math. Program. 85, 15–33 (1999)
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I–convex underestimating problems. Math. Program. 10, 147–175 (1976)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley Interscience, New York (1988)
Padberg, M.W., Roy, T.J.V., Wolsey, L.A.: Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842–861 (1985)
Rebennack, S., Nahapetyan, A., Pardalos, P.M.: Bilinear modeling solution approach for fixed charge network flow problems. Optim. Lett. 3, 347–355 (2009)
Richard, J.-P.P., Tawarmalani, M.: Lifting inequalities: a framework for generating strong cuts for nonlinear programs. Math. Program. 121, 61–104 (2010)
Sahinidis, N.V., Tawarmalani, M.: BARON. The Optimization Firm, LLC., Urbana-Champaign. Available at http://www.gams.com/dd/docs/solvers/baron.pdf (2005)
Sherali, H.D., Smith, J.C.: An improved linearization strategy for zero-one quadratic programming problems. Optim. Lett. 1, 33–47 (2007)
Smith, J.C., Lim, C.: Algorithms for network interdiction and fortification games. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds.) Pareto Optimality, Game Theory, and Equilibria, pp. 609–644. Springer, Berlin (2008)
Tawarmalani, M.: Inclusion certificates and simultaneous convexification of functions. Working paper (2012)
Tawarmalani, M., Richard, J.-P.P., Chung, K.: Strong valid inequalities for orthogonal disjunctions and polynomial covering sets. Technical Report 1213, Krannert School of Management, Purdue University (2008)
Tawarmalani, M., Richard, J.-P.P., Chung, K.: Strong valid inequalities for orthogonal disjunctions and bilinear covering sets. Math. Program. 124, 481–512 (2010)
Wolsey, L.A.: Faces for a linear inequality in \(0\mathord {-}1\) variables. Math. Program. 8, 165–178 (1975)
Wolsey, L.A.: Facets and strong valid inequalities for integer programs. Oper. Res. 24, 362–372 (1976)
Wolsey, L.A.: Valid inequalities and superadditivity for \(0\mathord {-}1\) integer programs. Math. Oper. Res. 2, 66–77 (1977)
Yaman, H.: The integer knapsack cover polyhedron. SIAM J. Discret. Math. 21, 551–572 (2007)
Ziegler, G.M.: Lectures on Polytopes. Springer, NY (1998)
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This work was supported by NSF CMMI Grants 0856605 and 0900065.
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Chung, K., Richard, JP.P. & Tawarmalani, M. Lifted inequalities for \(0\mathord {-}1\) mixed-integer bilinear covering sets. Math. Program. 145, 403–450 (2014). https://doi.org/10.1007/s10107-013-0652-1
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DOI: https://doi.org/10.1007/s10107-013-0652-1