The Hypervolume Newton Method for Constrained Multi-Objective Optimization Problems

Mathematical and Computational Applications, vol. 28, pp. 10

Abstract

Recently, the Hypervolume Newton Method (HVN) has been proposed as a fast and precise indicator-based method for solving unconstrained bi-objective optimization problems with objective functions. The HVN is defined on the space of (vectorized) fixed cardinality sets of decision space vectors for a given multi-objective optimization problem (MOP) and seeks to maximize the hypervolume indicator adopting the Newton–Raphson method for deterministic numerical optimization. To extend its scope to non-convex optimization problems, the HVN method was hybridized with a multi-objective evolutionary algorithm (MOEA), which resulted in a competitive solver for continuous unconstrained bi-objective optimization problems. In this paper, we extend the HVN to constrained MOPs with in principle any number of objectives. Similar to the original variant, the first- and second-order derivatives of the involved functions have to be given either analytically or numerically. We demonstrate the applicability of the extended HVN on a set of challenging benchmark problems and show that the new method can be readily applied to solve equality constraints with high precision and to some extent also inequalities. We finally use HVN as a local search engine within an MOEA and show the benefit of this hybrid method on several benchmark problems.

Authors 5

1. Hao Wang corresponding Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Leiden University, 2333 CA Leiden, The Netherlands

2. Michael Emmerich Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Leiden University, 2333 CA Leiden, The Netherlands

3. Andre H. Deutz Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Leiden University, 2333 CA Leiden, The Netherlands

4. Víctor Adrián Sosa Hernández

   Tecnológico de Monterrey

   Affiliation as printed

   School of Engineering and Sciences, Tecnológico de Monterrey, Av. Lago de Guadalupe Km 3.5, Atizapán de Zaragoza, Mexico City 52926, Mexico

5. Oliver Schütze

   Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional

   Affiliation as printed

   Computer Science Department, Cinvestav-IPN, Mexico City 07360, Mexico