A New Multiobjective Optimization Algorithm for Nonconvex Pareto Fronts and Objective Functions

Abstract

The main goal of a multi-objective optimization problem (MOP) is to construct the Pareto front (PF), which represents the best tradeoffs among the objectives to be optimized. The development of MOP methods has sought the generation of an even spread of Pareto optimal points as well as the treatment of the non-convexity on both the Pareto front and the functions to be minimized themselves.
Among other approaches, the normal boundary intersection (NBI) is able to deal with nonconvex PF and it also provides an even spread of Pareto optimal points. To deal with the non-convexity of the functions to be minimized some researchers have employed heuristic algorithms in the construction of the Pareto Front. However, the computational cost associated to heuristics is very high and the application of these methods to expensive functions (e.g. structures modeled by a finite element method) may become nonviable. Another difficulty arises in coupling a heuristic algorithm and the NBI approach: the latter imposes an equality constraint and the heuristic algorithm must then employ a penalty method to handle such a constraint, which usually worsens the performance of the algorithm. Clearly, the main issue for the utilization of heuristic algorithms is the computational cost they require. With the limitations of heuristic algorithms in mind, we propose a new algorithm for the solution of MOP whose PF and
the functions to be minimized are nonconvex. It applies the NBI to discretize the PF and it employs a global optimization algorithm based on a probabilistic restart procedure and local searches to minimize each NBI subproblem. The accuracy and efficiency of the proposed algorithm is shown in the numerical section comparing its results to well-known heuristic algorithms.