Non-linear Dimensionality Reduction Procedures for Certain Large-Dimensional Multi-objective Optimization Problems: Employing Correntropy and a Novel Maximum Variance Unfolding

Abstract

In our recent publication [1], we began with an understanding that many real-world applications of multi-objective optimization involve a large number (10 or more) of objectives but then, existing evolutionary multi-objective optimization (EMO) methods have primarily been applied to problems having smaller number of objectives (5 or less). After highlighting the major impediments in handling large number of objectives, we proposed a principal component analysis (PCA) based EMO procedure, for dimensionality reduction, whose efficacy was demonstrated by solving upto 50-objective optimization problems. Here, we are addressing the fact that, when the data points live on a non-linear manifold or that the data structure is non-gaussian, PCA which yields a smaller dimensional ’linear’ subspace may be ineffective in revealing the underlying dimensionality. To overcome this, we propose two new non-linear dimensionality reduction algorithms for evolutionary multi-objective optimization, namely C-PCA-NSGA-II and MVU-PCA-NSGA-II. While the former is based on the newly introduced correntropy PCA [2], the later implements maximum variance unfolding principle [3,4,5] in a novel way. We also establish the superiority of these new EMO procedures over the earlier PCA-based procedure, both in terms of accuracy and computational time, by solving upto 50-objective optimization problems.