Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

Abstract

We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, checking node fathoming, presolve, and duality gap measurement. Our branch-and-bound is predominantly a decision space search method because the branching is performed on the decision variables, akin to single objective problems, although we also sometimes split gaps and branch in the objective space. The various algorithms are implemented using a data structure for storing Pareto sets. Computational experiments are carried out on literature instances and on a new set of instances that we generate using a benchmark library (MIPLIB2017) for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm. Summary of Contribution: Biobjective mixed-integer optimization problems have two linear objectives and a mixed-integer feasible region. Such problems have many applications in operations research, because many real-world optimization problems naturally comprise two conflicting objectives to optimize or can be approximated in such a manner and are even harder than single objective mixed-integer programs. Solving them exactly requires the computation of all the nondominated solutions in the objective space, whereas some applications may also require finding at least one solution in the decision space corresponding to each nondominated solution. This paper provides an exact algorithm for solving these problems using the branch-and-bound method, which works predominantly in the decision space. Of the many ingredients of this algorithm, some parts are direct extensions of the single-objective version, but the main parts are newly designed algorithms to handle the distinct challenges of optimizing over two objectives. The goal of this study is to improve solution quality and speed and show that decision-space algorithms perform comparably to, and sometimes better than, algorithms that work mainly in the objective-space.

Enhancing Branch-and-Bound for Multi-Objective 0-1 Programming

“…In their paper, the authors use a surface as a lower bound set, namely the convex relaxation. Thereafter, the linear relaxation, weighted-sum scalarizations, and the linear relaxations of weighted sum scalarizations have been widely used in a similar way (see e.g., Vincent et al (2013), Stidsen et al (2014), Belotti et al (2016), Stidsen and Andersen (2018), Parragh and Tricoire (2019), Gadegaard et al (2019), Adelgren and Gupte (2022)). Although all these studies focus on the bi-objective case, the separating hypersurface principle from Sourd and Spanjaard (2008) is also applicable in higher dimensions.…”

“…Later, Vincent et al (2013) proposed a refined version of their framework for the bi-objective 0-1 case. The use of MOBB to solve bi-objective MOMILP was further studied by Belotti et al (2016) and Adelgren and Gupte (2022).…”

“…Recently, Adelgren and Gupte (2022) have proposed to use probing to enhance their bi-objective branchand-bound framework. The probing procedure of Adelgren and Gupte (2022) relies on solving the bi-objective linear relaxation based bound set, and showed promising results in their experiments. However, the impact of probing for problems with three or more objectives is unclear, as bound sets are more complex to compute.…”

“…Alternatively, a MOILP can be solved using a Decision Space Search (DSS) algorithm, typically a Multi-Objective Branch & Bound (MOBB) algorithm. Almost all recent contributions in this area address the bi-objective case (Adelgren and Gupte, 2022;Gadegaard et al, 2019;Parragh and Tricoire, 2019;Stidsen et al, 2014), and they all rely on an efficient branching scheme that creates sub-problems in the objective space. Forget et al (2020b) have recently generalized this scheme to problems with more than two objectives.…”

The bi-objective multimodal car-sharing problem

“…These so called criterion space methods, embed a single-objective optimization problem and systematically enumerate the Pareto frontier. However, recent works focus on adapting the branch-and-bound algorithm to solve the multi-objective case in a single run [54,61,47,2]. A recent overview of exact methods for multi-objective optimization is provided in Ehrgott et al [23].…”