Bi-objective optimisation over a set of convex sub-problems

During the last decades, research in multi-objective optimisation has seen considerable growth. However, this activity has been focused on linear, non-linear, and combinatorial optimisation with multiple objectives. Multi-objective mixed integer (linear or non-linear) programming has received considerably less attention. In this paper we propose an algorithm to compute a finite set of non-dominated points/efficient solutions of a bi-objective mixed binary optimisation problems for which the sub-problems obtained when fixing the binary variables are convex, and there is a finite set of feasible binary variable vectors. Our method uses bound sets and exploits the convexity property of the sub-problems to find a set of efficient solutions for the main problem. Our algorithm creates and iteratively updates bounds for each vector in the set of feasible binary variable vectors, and uses these bounds to guarantee that a set of exact non-dominated points is generated. For instances where the set of feasible binary variable vectors is too large to generate such provably optimal solutions within a reasonable time, our approach can be used as a matheuristic by heuristically selecting a promising subset of binary variable vectors to explore. This investigation is motivated by the problem of beam angle optimisation arising in radiation therapy planning, which we solve heuristically to provide numerical results.