A general framework for evolutionary multiobjective optimization via manifold learning

Abstract

Under certain mild condition, the Pareto-optimal set (PS) of a continuous multiobjective optimization problem, with m objectives, is a piece-wise continuous (m−1)-dimensional manifold. This regularity property is important, yet has been unfortunately ignored in many evolutionary multiobjective optimization (EMO) studies. The first work that explicitly takes advantages of this regularity property in EMO is the regularity model-based multiobjective estimation of distribution algorithm (RM-MEDA). However, its performance largely depends on its model parameter, which is problem dependent. Manifold learning, also known as nonlinear dimensionality reduction, is able to discover the geometric property of a low-dimensional manifold embedded in the high-dimensional ambient space. This paper presents a general framework that applies advanced manifold learning techniques in EMO. At each generation, we first use a principal curve algorithm to obtain an approximation of the PS manifold. Then, the Laplacian eigenmaps algorithm is employed to find the low-dimensional representation of this PS approximation. Afterwards, we identify the neighborhood relationship in this low-dimensional representation, which is also applicable for the original high-dimensional data. Based on the neighborhood relationship, we can interpolate new candidate solutions that obey the geometric property of the PS manifold. Empirical results validate the effectiveness of our proposal.