A model of anytime algorithm performance for bi-objective optimization

Anytime algorithms allow a practitioner to trade-off runtime for solution quality. This is of particular interest in multi-objective combinatorial optimization since it can be infeasible to identify all efficient solutions in a reasonable amount of time. We present a theoretical model that, under some mild assumptions, characterizes the “optimal” trade-off between runtime and solution quality, measured in terms of relative hypervolume, of anytime algorithms for bi-objective optimization. In particular, we assume that efficient solutions are collected sequentially such that the collected solution at each iteration maximizes the hypervolume indicator, and that the non-dominated set can be well approximated by a quadrant of a superellipse. We validate our model against an “optimal” model that has complete knowledge of the non-dominated set. The empirical results suggest that our theoretical model approximates the behavior of this optimal model quite well. We also analyze the anytime behavior of an ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-constraint algorithm, and show that our model can be used to guide the algorithm and improve its anytime behavior.