Bi-objective optimisation over a set of convex sub-problems

During the last decades, research in multi-objective optimisation has seen considerable growth. However, this activity has been focused on linear, non-linear, and combinatorial optimisation with multiple objectives. Multi-objective mixed integer (linear or non-linear) programming has received consid...

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Vydáno v:	Annals of operations research Ročník 319; číslo 2; s. 1507 - 1532
Hlavní autoři:	Cabrera-Guerrero, Guillermo, Ehrgott, Matthias, Mason, Andrew J., Raith, Andrea
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.12.2022 Springer Springer Nature B.V
Témata:	Algorithms Business and Management Combinatorial analysis Combinatorics Convex domains Convexity Heuristic Integer programming Linear programming Mathematical optimization Mixed integer Multiple objective analysis Objectives Operations research Operations Research/Decision Theory Optimization Original Research Radiation therapy Theory of Computation Variables Chile Mixed binary programming Intensity modulated radiation therapy Convex optimisation Multiple objective programming
ISSN:	0254-5330, 1572-9338
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Abstract	During the last decades, research in multi-objective optimisation has seen considerable growth. However, this activity has been focused on linear, non-linear, and combinatorial optimisation with multiple objectives. Multi-objective mixed integer (linear or non-linear) programming has received considerably less attention. In this paper we propose an algorithm to compute a finite set of non-dominated points/efficient solutions of a bi-objective mixed binary optimisation problems for which the sub-problems obtained when fixing the binary variables are convex, and there is a finite set of feasible binary variable vectors. Our method uses bound sets and exploits the convexity property of the sub-problems to find a set of efficient solutions for the main problem. Our algorithm creates and iteratively updates bounds for each vector in the set of feasible binary variable vectors, and uses these bounds to guarantee that a set of exact non-dominated points is generated. For instances where the set of feasible binary variable vectors is too large to generate such provably optimal solutions within a reasonable time, our approach can be used as a matheuristic by heuristically selecting a promising subset of binary variable vectors to explore. This investigation is motivated by the problem of beam angle optimisation arising in radiation therapy planning, which we solve heuristically to provide numerical results.
AbstractList	During the last decades, research in multi-objective optimisation has seen considerable growth. However, this activity has been focused on linear, non-linear, and combinatorial optimisation with multiple objectives. Multi-objective mixed integer (linear or non-linear) programming has received considerably less attention. In this paper we propose an algorithm to compute a finite set of non-dominated points/efficient solutions of a bi-objective mixed binary optimisation problems for which the sub-problems obtained when fixing the binary variables are convex, and there is a finite set of feasible binary variable vectors. Our method uses bound sets and exploits the convexity property of the sub-problems to find a set of efficient solutions for the main problem. Our algorithm creates and iteratively updates bounds for each vector in the set of feasible binary variable vectors, and uses these bounds to guarantee that a set of exact non-dominated points is generated. For instances where the set of feasible binary variable vectors is too large to generate such provably optimal solutions within a reasonable time, our approach can be used as a matheuristic by heuristically selecting a promising subset of binary variable vectors to explore. This investigation is motivated by the problem of beam angle optimisation arising in radiation therapy planning, which we solve heuristically to provide numerical results.
Audience	Academic
Author	Mason, Andrew J. Cabrera-Guerrero, Guillermo Ehrgott, Matthias Raith, Andrea
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SubjectTerms	Algorithms Business and Management Combinatorial analysis Combinatorics Convex domains Convexity Heuristic Integer programming Linear programming Mathematical optimization Mixed integer Multiple objective analysis Objectives Operations research Operations Research/Decision Theory Optimization Original Research Radiation therapy Theory of Computation Variables
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Title	Bi-objective optimisation over a set of convex sub-problems
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