A deterministic optimization algorithm for nonconvex and combinatorial bi-objective programming

Abstract: any practical multiobjective optimization (MOO) problems include discrete decision variables and/or nonlinear model equations and exhibit disconnected or smooth but nonconvex Pareto surfaces. Scalarization methods, such as the weighted-sum and sandwich (SD) algorithms, are common approaches to solving MOO problems but may fail on nonconvex or discontinuous Pareto fronts. In the current work, motivated by the well-known normal boundary intersection (NBI) method and the SD algorithm, we present SDNBI, a new algorithm for bi-objective optimization (BOO) designed to address the theoretical and numerical challenges associated with the reliable solution of general nonconvex and discrete BOO problems. The main improvements in the algorithm are the effective exploration of the nonconvex regions of the Pareto front and, uniquely, the early identification of regions where no additional Pareto solutions exist. The performance of the SDNBI algorithm is assessed based on the accuracy of the approximation of the Pareto front constructed over the disconnected nonconvex objective domains. The new algorithm is compared with two MOO approaches, the modified NBI method and the SD algorithm, using published benchmark problems. The results indicate that the SDNBI algorithm outperforms the modified NBI and SD algorithms in solving convex, nonconvex-continuous, and combinatorial problems, both in terms of computational cost and of the overall quality of the Pareto-optimal set, suggesting that the SDNBI algorithm is a promising alternative for solving BOO problems.