A Branch and Bound Algorithm for Multiobjective Optimization Problems Using General Ordering Cones

TL;DR

This paper introduces a novel branch and bound algorithm for multiobjective optimization that uses general ordering cones to efficiently approximate parts of the Pareto set, reducing computational costs.

Contribution

It proposes a new cone dominance-based discarding test that incorporates preference information via general cones, extending traditional Pareto dominance methods.

Findings

Algorithm converges globally.

Effective on test instances and real-world problems.

Reduces computational effort in Pareto set approximation.

Abstract

Many existing branch and bound algorithms for multiobjective optimization problems require a significant computational cost to approximate the entire Pareto optimal solution set. In this paper, we propose a new branch and bound algorithm that approximates a part of the Pareto optimal solution set by introducing the additional preference information in the form of ordering cones. The basic idea is to replace the Pareto dominance induced by the nonnegative orthant with the cone dominance induced by a larger ordering cone in the discarding test. In particular, we consider both polyhedral and non-polyhedral cones, and propose the corresponding cone dominance-based discarding tests, respectively. In this way, the subboxes that do not contain efficient solutions with respect to the ordering cone will be removed, even though they may contain Pareto optimal solutions. We prove the global convergence of the proposed algorithm. Finally, the proposed algorithm is applied to a number of test instances as well as to 2- to 5-objective real-world constrained problems.