Extremized PICEA-g for Nadir Point Estimation in Many-Objective Optimization

Abstract

Nadir point, constructed by the worst Pareto optimal objective values, plays an important role in multi-objective optimization and decision making. For example, the nadir point is often a pre-requisite in many multi-criterion decision making approaches. Along with the ideal point, the nadir point can be applied to normalize solutions so as to facilitate a comparison and aggregation of objectives. Moreover, nadir point is useful in visualization software catered for multi-objective optimization. However, the estimation of nadir point is still a challenging problem, particularly, for optimization and/or decision-making problems with many objectives. In this paper, a modified preference-inspired coevolutionary algorithm using goal vectors (PICEA-g) called extremized PICEA-g is proposed to estimate the nadir point. The extremized PICEA-g, denoted as e-PICEA-g, is an $\left( {N + N} \right)$ elitist algorithm and employs a two-phase selection strategy. In the first-phase $\left( {N + K} \right)$ solutions are selected out from the overall $2N$ solutions based on the dominance-level and an angle based closeness indicator. In the second-phase the selected $\left( {N + K} \right)$ solutions are further filtered by removing $K$ poor ones in terms of their fitness calculated by a slightly modified PICEA-g fitness scheme. By the two-phase selection strategy, the e-PICEA-g skillfully harnesses the advantages of edge-point-to-nadir and extreme-to-nadir principles. Experimental results demonstrate the efficiency and effectiveness of the e-PICEA-g on many-objective optimization benchmarks with up to 13 objectives.