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.