A Line Complex-Based Evolutionary Algorithm for Many-Objective Optimization

In solving many-objective optimization problems (MaOPs), existing nondominated sorting-based multi-objective evolutionary algorithms suffer from the fast loss of selection pressure. Most candidate solutions become nondominated during the evolutionary process, thus leading to the failure of producing off-spring toward Pareto-optimal front with diversity. Can we find a more effective way to select nondominated solutions and resolve this issue? To answer this critical question, this work proposes to evolve solutions through line complex rather than solution points in Euclidean space. First, Plücker coordinates are used to project solution points to line complex composed of position vectors and momentum ones. Besides position vectors of the solution points, momentum vectors are used to extend the comparability of non-dominated solutions and enhance selection pressure. Then, a new distance function designed for high-dimensional space is proposed to replace Euclidean distance as a more effective distance-based estimator. Based on them, a novel many-objective evolutionary algorithm (MaOEA) is proposed by integrating a line complex-based environmental selection strategy into the NSGA-III framework. The proposed algorithm is compared with the state of the art on widely used benchmark problems with up to 15 objectives. Experimental results demonstrate its superior competitiveness in solving MaOPs.

I. Introduction

Multi-objective optimization problems (MOPs) exist widely in practical applications [1], [2]. They have received great attention in the system engineering community. MOPs can be defined as follows: $$\begin{align*} &\text{minimize}\ \ F(\boldsymbol{x})=(f_{1}(\boldsymbol{x}), f_{2}(\boldsymbol{x}), \ldots, f_{M}(\boldsymbol{x}))\in \mathbb{F}\\ &\mathrm{s}.\!\mathrm{t}.\quad \boldsymbol{x}\in \boldsymbol{X}\tag{1} \end{align*}$$ where is a decision vector with decision variables, refers to the decision space. is the objective vector consisting of objective functions denotes the objective space. Generally, the MOPs with more than three objectives () are regarded as many-objective optimization problems (MaOPs).