Decomposition in Decision and Objective Space for Multi-Modal Multi-Objective Optimization

Multi-modal multi-objective optimization problems (MMMOPs) have multiple solution vectors mapping to the same objective vector. For MMMOPs, it is important to discover equivalent solutions associated with each point in the Pareto-Front for allowing end-users to make informed decisions. Prevalent multi-objective evolutionary algorithms are incapable of searching for multiple solution subsets, whereas, algorithms designed for MMMOPs demonstrate degraded performance in the objective space. This motivates the design of better algorithms for addressing MMMOPs. The present work highlights the disadvantage of using crowding distance in the decision space when solving MMMOPs. Subsequently, an evolutionary framework, called graph Laplacian based Optimization using Reference vector assisted Decomposition (LORD), is proposed, which is the first algorithm to use decomposition in both objective and decision space for dealing with MMMOPs. Its filtering step is further extended to present LORD-II algorithm, which demonstrates its dynamics on multi-modal many-objective problems. The efficacy of the frameworks are established by comparing their performance on 34 test instances (obtained from the CEC 2019 multi-modal multi-objective test suite) with the state-of-the-art algorithms for MMMOPs and other multi- and many-objective evolutionary algorithms. The manuscript is concluded mentioning the limitations of the proposed frameworks and future directions to design still better algorithms for MMMOPs.