We study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints. These models reduce to a mixed-integer linear program (MILP) when the concave constraints are ignored, or to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms designed to solve these problems to global optimality, cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the MILP only. We propose a technique to obtain valid inequalities that are based on both the MILP constraints and the concave constraints. We begin by characterizing the convex hull of a four-dimensional set consisting of a single binary indicator variable, a single concave constraint, and two linear inequalities. Using this analysis, we demonstrate how valid inequalities for the single node flow set and for the lot-sizing polyhedron can be “tilted” to give valid inequalities that also account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot-sizing problems with concave costs. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed-integer nonlinear programs.
Mathematics Subject Classification
This is a preview of subscription content, log in to check access.
The work was supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Contract Number DE-AC02-06CH11357.
Berthold, T., Heinz, S., Vigerske, S.: Extending a CIP framework to solve MIQCPs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and Its Applications, vol. 154, pp. 427–444. Springer, Berlin (2012)CrossRefGoogle Scholar
Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: Mixed integer nonlinear programming, pp. 1–39. Springer, Berlin (2012)Google Scholar
Borghetti, A., D’Ambrosio, C., Lodi, A., Martello, S.: An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir. IEEE Trans. Power Syst. 23(3), 1115–1124 (2008)CrossRefGoogle Scholar
D’Ambrosio, C., Lodi, A., Wiese, S., Bragalli, C.: Mathematical programming techniques in water network optimization. Eur. J. Oper. Res. 243(3), 774788 (2015)MathSciNetMATHGoogle Scholar
D’Ambrosio, C., Lee, J., Wächter, A.: An algorithmic framework for MINLP with separable non-convexity. In: Lee, J., Leyffer, S. (eds.) Nonlinear Optimization: Algorithmic Advances and Applications, IMA Volumes in Mathematics and its Applications, vol. 154, pp. 315–347 (2012)Google Scholar
Humpola, J., Fügenschuh, A.: A new class of valid inequalities for nonlinear network design problems. Technical Report 13-06, ZIB, Konrad-Zuse-Zentrum für Informationstechnik Berlin (2013)Google Scholar
Lim, C.H., Linderoth, J., Luedtke, J.: Valid inequalities for separable concave constraints with indicator variables. In: IPCO 2016: The Sixteenth Conference on Integer Programming and Combinatorial Optimization, vol. 9682, pp. 275–286. Springer, Berlin (2016)Google Scholar
Martin, A., Möller, M., Moritz, S.: Mixed integer models for the stationary case of gas network optimization. Math. Program. 105(2), 563–582 (2006)MathSciNetCrossRefMATHGoogle Scholar
Misener, R., Floudas, C.A.: Antigone: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59, 503–526 (2014)MathSciNetCrossRefMATHGoogle Scholar