LIBEA: A Lebesgue Indicator-Based Evolutionary Algorithm for multi-objective optimization

Abstract

Abstract In multi-objective optimization, the hypervolume indicator (also known as S metric or Lebesgue measure) evaluates, at the same time, the approximation and distribution of non-dominated solutions along the Pareto optimal front of a multi-objective optimization problem (MOP). This indicator has become one of the most studied performance measures in the so-called indicator-based evolutionary algorithms (IBEAs). However, IBEAs based on hypervolume are limited by their high computational cost which increases with the number of objectives. As a consequence, several researchers have proposed new strategies to reduce the computational time of IBEAs based on this indicator. In this regard, the present paper introduces a Lebesgue indicator-based evolutionary algorithm (LIBEA) for continuous box-constrained multi-objective optimization problems. In contrast to most of IBEAs based on hypervolume, LIBEA implicitly employs the regularity property of continuous MOPs which has been suggested to solve continuous MOPs with difficult properties. The proposed LIBEA is validated on a set of test functions with different properties regarding separability, multi-modality, and different Pareto front geometries including convexity, concavity, and discontinuity. The performance of LIBEA is compared against three state-of-the-art evolutionary multi-objective algorithms (EMOAs) based on different principles. For a more comprehensive study, we evaluate the performance of our proposed approach and the adopted EMOAs over two real-world problems whose properties are unknown. We show that our proposed approach is highly competitive and that, in many cases, LIBEA significantly improved the state-of-the-art EMOAs on the test problems adopted in our comparative study.