Branch-and-bound and objective branching with three or more objectives

Forget, Nicolas; Gadegaard, Sune Lauth; Klamroth, Kathrin; Nielsen, Lars Relund; Przybylski, Anthony

doi:10.1016/j.cor.2022.106012

Cited by 15 publications

(8 citation statements)

References 29 publications

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“…This also includes a strengthened version of Pareto branching/image space branching (Parragh & Tricoire, 2019; Stidsen et al, 2014). This branching strategy is also discussed by Forget et al (2020), where a general branch‐and‐bound framework and a generalization of Pareto branching/image space branching to more than two objectives is presented for multiobjective combinatorial problems.…”

Section: Algorithms For Multiobjective Integer Optimization Problemsmentioning

confidence: 98%

“…The latter branching strategy comes into play, when the current image is dominated but cannot be fathomed. Then, in the branching step new bounds on the objective function value are installed (see also Forget et al, 2020). Additionally, further improvements like image space slicing are presented (see Stidsen & Andersen, 2018 in Section 4.3).…”

Section: Algorithms For Multiobjective Mixed‐integer Optimization Pro...mentioning

confidence: 99%

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Exact algorithms for multiobjective linear optimization problems with integer variables: A state of the art survey

Halffmann

Schäfer

Dächert

et al. 2022

Multi Criteria Decision Anal

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We provide a comprehensive overview of the literature of algorithmic approaches for multiobjective mixed-integer and integer linear optimization problems. More precisely, we categorize and display exact methods for multiobjective linear problems with integer variables for computing the entire set of nondominated images. Our review lists 108 articles and is intended to serve as a reference for all researchers who are familiar with basic concepts of multiobjective optimization and who have an interest in getting a thorough view on the state-of-the-art in multiobjective mixedinteger programming.

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Section: Algorithms For Multiobjective Integer Optimization Problemsmentioning

confidence: 98%

Section: Algorithms For Multiobjective Mixed‐integer Optimization Pro...mentioning

confidence: 99%

Exact algorithms for multiobjective linear optimization problems with integer variables: A state of the art survey

Halffmann

Schäfer

Dächert

et al. 2022

Multi Criteria Decision Anal

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show abstract

“…Later, Adelgren and Gupte (2022) also showed promising results using objective branching in MOBB applied to bi-objective MOMILP. Forget et al (2020a) extended objective branching to the multi-objective case. In their paper, the authors highlighted several challenges that arise when three or more objective functions are considered.…”

Section: Related Workmentioning

confidence: 99%

“…To do so, objective branching is used first. This technique was initially introduced for the bi-objective case by Stidsen et al (2014), further improved by Gadegaard et al (2019) and Parragh and Tricoire (2019), and finally extended to the multi-objective case in Forget et al (2020a). It consists of creating disjoint sub-problems in the objective space when the lower bound set is partially dominated by the upper bound set, with the purpose of discarding regions that cannot contain any new non-dominated points.…”

Section: Branch-and-bound Frameworkmentioning

confidence: 99%

“…In the example from Figure 1, this results in three subproblems, defined by three super local upper bounds depicted by the large circles. Recently, Forget et al (2020a) showed that the computation of the subproblems in the case where 𝑝 ≥ 3 was more complex but still possible. However, the increased complexity resulted in a less significant benefit of using objective branching when 𝑝 ≥ 3 compared to the case where 𝑝 = 2.…”

Section: Objective Branching Induced Enhancementsmentioning

confidence: 99%

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

Forget¹,

Parragh²

2022

Preprint

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In the bi-objective branch-and-bound literature, a key ingredient is objective branching, i.e. to create smaller and disjoint sub-problems in the objective space, obtained from the partial dominance of the lower bound set by the upper bound set. When considering three or more objective functions, however, applying objective branching becomes more complex, and its benefit has so far been unclear. In this paper, we investigate several ingredients which allow to better exploit objective branching in a multi-objective setting. We extend the idea of probing to multiple objectives, enhance it in several ways, and show that when coupled with objective branching, it results in significant speed-ups in terms of CPU times. We also investigate cut generation based on the objective branching constraints. Besides, we generalize the best-bound idea for node selection to multiple objectives and we show that the proposed rules outperform the, in the multi-objective literature, commonly employed depth-first and breadth-first strategies. We also analyze problem specific branching rules. We test the proposed ideas on available benchmark instances for three problem classes with three and four objectives, namely the capacitated facility location problem, the uncapacitated facility location problem, and the knapsack problem. Our enhanced multi-objective branch-and-bound algorithm outperforms the best existing branch-and-bound based approach and is the first to obtain competitive and even slightly better results than a state-of-the-art objective space search method on a subset of the problem classes.

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An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems

Bökler,

Parragh,

Sinnl

et al. 2024

Math Meth Oper Res

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In this paper, we present an outer approximation algorithm for computing the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems (MOMILPs). It produces the extreme points (i.e., the vertices) as well as the facets of the Edgeworth–Pareto hull. We note that these extreme points are the extreme supported non-dominated points of a MOMILP. We also show how to extend the concept of geometric duality for multi-objective linear programming problems to the Edgeworth–Pareto hull of MOMILPs and use this extension to develop the algorithm. The algorithm relies on a novel oracle that solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the extreme supported non-dominated points in the case of MOMILPs. Thus, for MOMILPs for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay—a result that was formerly only known for the computation of extreme supported non-dominated points. Our algorithm can be an attractive option to compute lower bound sets within multi-objective branch-and-bound algorithms for solving MOMILPs. This is for several reasons as (i) the algorithm starts from a trivial valid lower bound set then iteratively improves it, thus at any iteration of the algorithm a lower bound set is available; (ii) the algorithm also produces efficient solutions (i.e., solutions in the decision space); (iii) in any iteration of the algorithm, a relaxation of the MOMILP can be solved, and the obtained points and facets still provide a valid lower bound set. Moreover, for the special case of multi-objective linear programming problems, the algorithm solves the problem to global optimality. A computational study on a set of benchmark instances from the literature is provided.

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Branch-and-bound and objective branching with three or more objectives

Cited by 15 publications

References 29 publications

Exact algorithms for multiobjective linear optimization problems with integer variables: A state of the art survey

Exact algorithms for multiobjective linear optimization problems with integer variables: A state of the art survey

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

An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems

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