Computing Optimality Certificates for Convex Mixed-Integer Nonlinear Problems

Abstract

Every optimization problem has a corresponding verification problem that checks whether a given optimal solution is in fact optimal. In the literature, there are a lot of such ways to verify optimality for a given solution, for example, the branch-and-bound tree. To simplify this task, optimality certificates were introduced for convex mixed-integer nonlinear programs, and it was shown that the sizes of the certificates are bounded in terms of the number of integer variables. We introduce an algorithm to compute the certificates and conduct computational experiments. Through the experiments, we show that the optimality certificates can be surprisingly small. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [CRC 154 Subproject A05, CRC 154 Subproject B07, and SFB Transregio 154], the Bundesministerium für Wirtschaft und Energie. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0099 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0099 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .