This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the focus of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space which was already used for continuous nonconvex multiobjective optimization. We discuss how this basic method can be combined with appropriate branching rules and lower bounding procedures such that it is also implementable and convergent for multiobjective mixed-integer optimization problems. For improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a deterministic solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare with existing deterministic procedures.
Optimization Online, Preprint ID 2022-02-8796.