Strengthening the sequential convex MINLP technique by perspective reformulations
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The sequential convex MINLP (SC-MINLP) technique is a solution method for nonconvex mixed-integer nonlinear problems (MINLPs) where the nonconvexities are separable. It is based on solving a sequence of convex MINLPs which trade a better and better relaxation of the nonconvex part of the problem with the introduction of more and more piecewise-linear nonconvex terms, and therefore binary variables. The convex MINLPs are obtained by partitioning the domain of each separable nonconvex term in the intervals in which it is convex and those in which it is concave. In the former, the term is left in its original form, while in the latter it is piecewise-linearized. Since each interval corresponds to a semi-continuous variable, we propose to modify the convex terms using the Perspective Reformulation technique to strengthen the bounds. We show by means of experimental results on different classes of instances that doing so significantly decreases the solution time of the convex MINLPs, which is the most time consuming part of the approach, and has therefore the potential to improving the overall effectiveness of SC-MINLP.
KeywordsGlobal optimization Nonconvex separable functions Sequential convex MINLP technique Perspective reformulation
The second and the third authors are partially supported by the Project MIUR-PRIN 2015B5F27W. This paper has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 764759.
- 1.Bonami, P., Lee, J.: Bonmin users’ manual. Technical Report (June 2006)Google Scholar
- 3.D’Ambrosio, C., Lee, J., Wächter, A.: A global-optimization algorithm for mixed-integer nonlinear programs having separable non-convexity. In: Algorithms—ESA 2009. LNCS, vol. 5757, pp. 107–118. Springer, Berlin (2009)Google Scholar
- 10.Gleixner, A., Eifler, L., Gally, T., Gamrath, G., Gemander, P., Gottwald, R.L., Hendel, G., Hojny, C., Koch, T., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schlösser, F., Serrano, F., Shinano, Y., Viernickel, J.M., Vigerske, S., Weninger, D., Witt, J.T., Witzig, J.: The SCIP optimization suite 5.0. Technical Report 17-61, ZIB, Takustr.7, 14195 Berlin (2017)Google Scholar
- 11.IBM: ILOG CPLEX 12.7 User’s Manual. IBM (2016)Google Scholar
- 12.Mahajan, A., Leyffer, S., Linderoth, J., Luedtke, J., Munson, T.: Minotaur: A mixed-integer nonlinear optimization toolkit. Optimization Online 6275 (2017)Google Scholar