An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems
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 extrem...
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| Veröffentlicht in: | Mathematical methods of operations research (Heidelberg, Germany) Jg. 100; H. 1; S. 263 - 290 |
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01.08.2024
Springer Berlin Heidelberg Springer Nature B.V |
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| Abstract | 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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| AbstractList | 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. |
| Author | Sinnl, Markus Parragh, Sophie N Tricoire, Fabien Bökler, Fritz |
| Author_xml | – sequence: 1 givenname: Fritz surname: Bökler fullname: Bökler, Fritz – sequence: 2 givenname: Sophie N surname: Parragh fullname: Parragh, Sophie N – sequence: 3 givenname: Markus surname: Sinnl fullname: Sinnl, Markus – sequence: 4 givenname: Fabien surname: Tricoire fullname: Tricoire, Fabien |
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| CitedBy_id | crossref_primary_10_1007_s00186_024_00874_z crossref_primary_10_1007_s10898_025_01512_6 crossref_primary_10_1016_j_ejco_2024_100099 crossref_primary_10_1080_0305215X_2025_2464867 |
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| SubjectTerms | Algorithms Apexes Approximation Business and Management Calculus of Variations and Optimal Control; Optimization Convexity Integer programming Linear programming Lower bounds Mathematics Mathematics and Statistics Mixed integer Mixed-integer programming Multi-objective optimization Multiple objective analysis Operations Research/Decision Theory Original Article Outer approximation Polynomials Simplex method |
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| Title | An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems |
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