The Hypervolume Newton Method for Constrained Multi-objective Optimization Problems

Preprints.org

Abstract

Recently, the Hypervolume Newton method (HVN) has been proposed as fast and precise indicator-based method for solving unconstrained bi-objective optimization problems with objective functions that are at least twice continuously differentiable. 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. We demonstrate the applicability of the extended HVN on a set of challenging benchmark problems and show that the new method can be readily be applied to solve equality constraints with a high precision problems, and to some extend also inequalities. We finally use HVN as local search engine within a 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 , The Netherlands ;

   Leiden Institute of Advanced Computer Science, Leiden University, 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 , The Netherlands ;

   Leiden Institute of Advanced Computer Science, Leiden University, 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 , The Netherlands ;

   Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands;

4. Víctor Adrián Sosa Hernández

   Tecnológico de Monterrey

   Affiliation as printed

   Tecnologico de Monterrey , School of Engineering and Sciences , Mexico ;

5. Oliver Schütze

   Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional

   Affiliation as printed

   Computer Science Department , Cinvestav-IPN , Mexico;

   Computer Science Department, Cinvestav-IPN, Mexico;