Methods for multi-objective optimization: An analysis

•Given some prior information we can find the optimal scalarizing function.•No theoretical evidence to support the superiority of decomposition methods.•Constant direction of search could be the answer to faster convergence.•Pareto-dominance and the Chebyshev scalarizing function are equivalent.

Decomposition-based methods are often cited as the solution to multi-objective nonconvex optimization problems with an increased number of objectives. These methods employ a scalarizing function to reduce the multi-objective problem into a set of single objective problems, which upon solution yield a good approximation of the set of optimal solutions. This set is commonly referred to as Pareto front. In this work we explore the implications of using decomposition-based methods over Pareto-based methods on algorithm convergence from a probabilistic point of view. Namely, we investigate whether there is an advantage of using a decomposition-based method, for example using the Chebyshev scalarizing function, over Pareto-based methods. We find that, under mild conditions on the objective function, the Chebyshev scalarizing function has an almost identical effect to Pareto-dominance relations when we consider the probability of finding superior solutions for algorithms that follow a balanced trajectory. We propose the hypothesis that this seemingly contradicting result compared with currently available empirical evidence, signals that the disparity in performance between Pareto-based and decomposition-based methods is due to the inability of the former class of algorithms to follow a balanced trajectory. We also link generalized decomposition to the results in this work and show how to obtain optimal scalarizing functions for a given problem, subject to prior assumptions on the Pareto front geometry.