On the Performance Degradation of Dominance-Based Evolutionary Algorithms in Many-Objective Optimization

In the last decade, it has become apparent that the performance of Pareto-dominance-based evolutionary multiobjective optimization algorithms degrades as the number of objective functions of the problem, given by <inline-formula> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula>, grows. This performance degradation has been the subject of several studies in the last years, but the exact mechanism behind this phenomenon has not been fully understood yet. This paper presents an analytical study of this phenomenon under problems with continuous variables, by a simple setup of quadratic objective functions with spherical contour curves and a symmetrical arrangement of the function minima location. Within such a setup, some analytical formulas are derived to describe the probability of the optimization progress as a function of the distance <inline-formula> <tex-math notation="LaTeX">${\lambda }$ </tex-math></inline-formula> to the exact Pareto-set. A main conclusion is stated about the nature and structure of the performance degradation phenomenon in many-objective problems: when a current solution reaches a <inline-formula> <tex-math notation="LaTeX">${\lambda }$ </tex-math></inline-formula> that is an order of magnitude smaller than the length of the Pareto-set, the probability of finding a new point that dominates the current one is given by a power law function of <inline-formula> <tex-math notation="LaTeX">${\lambda }$ </tex-math></inline-formula> with exponent <inline-formula> <tex-math notation="LaTeX">${(n-1)}$ </tex-math></inline-formula>. The dimension of the space of decision variables has no influence on that exponent. Those results give support to a discussion about some general directions that are currently under consideration within the research community.