A branch-and-bound based heuristic algorithm for convex multi-objective MINLPs

We study convex multi-objective Mixed Integer Non-Linear Programming problems (MINLPs), which are characterized by multiple objective functions and non linearities, features that appear in real-world applications. To derive a good approximated set of non-dominated points for convex multi-objective MINLPs, we propose a heuristic based on a branch-and-bound algorithm. It starts with a set of feasible points, obtained, at the root node of the enumeration tree, by iteratively solving, with an ε-constraint method, a single objective model that incorporates the other objective functions as constraints. Lower bounds are derived by optimally solving Non-Linear Programming problems (NLPs). Each leaf node of the enumeration tree corresponds to a convex multi-objective NLP, which is solved heuristically by varying the weights in a weighted sum approach. In order to improve the obtained points and remove dominated ones, a tailored refinement procedure is designed. Although the proposed method makes no assumptions on the number of objective functions nor on the type of the variables, we test it on bi-objective mixed binary problems. In particular, as a proof-of-concept, we tested the proposed heuristic algorithm on instances of a real-world application concerning power generation, and instances of the convex biobjective Non-Linear Knapsack Problem. We compared the obtained results with those derived by well-known scalarization methods, showing the effectiveness of the proposed method.