Properties of enclosures in multiobjective optimization

Abstract

A widely used approximation concept in multiobjective optimization is the concept of enclosures. These are unions of boxes defined by lower and upper bound sets that are used to cover optimal sets of multiobjective optimization problems in the image space. The width of an enclosure is taken as a quality measure. In this paper, we provide properties of enclosures and their width in multiobjective optimization. To apply enclosures for warm-start strategies and for approximations of optimal sets, we discuss under which conditions enclosures have nonempty interior and whether they coincide with the closure of their interior. We extend the optimality concepts of $\epsilon$-minimality and $\epsilon$-weak minimality from multiobjective optimization and introduce new optimality concepts for multiobjective optimization caused by relaxations of multiobjective optimization problems. We show that the enclosures and their widths are suitable for determining these new optimal points. We provide some calculation and estimation rules for the width of enclosures, such as a monotonicity, decomposition, and combination property and a triangular inequality-like relation. These are important for convergence examinations for approximation algorithms. All our examinations are driven by the requirements to further explore enclosure-based algorithms for the numerical computation of optimal points in multiobjective optimization. Hence, this paper provides a toolbox of theoretical results for enclosures that supports the development of convergence proofs of image-space-based approximation methods for several classes of multiobjective optimization problems.