Article

Journal of Global Optimization

, Volume 50, Issue 3, pp 397-416

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

An approximation algorithm for convex multi-objective programming problems

  • Matthias EhrgottAffiliated withDepartment of Engineering Science, The University of Auckland
  • , Lizhen ShaoAffiliated withSchool of Information Engineering, University of Science and Technology Beijing Email author 
  • , Anita SchöbelAffiliated withFakultät für Mathematik, Georg-August Universität Göttingen

Abstract

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson’s outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.

Keywords

Multi-objective optimization Convex optimization Approximation algorithm ε-nondominated point