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

Cabrera‐Guerrero, Guillermo; Ehrgott, Matthias; Mason, A. James; Raith, Andrea

doi:10.1007/s10479-020-03910-3

Cited by 10 publications

(8 citation statements)

References 45 publications

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“…As mentioned in Section 2, the gEUD-based model in Equation ( 6) is considered in this paper to solve the MO-FMO problem in Equation (1). This model has been previously used in [2,3,8,9,16].…”

Section: Geud-based Mo-bao: Mathematical Formulationmentioning

confidence: 99%

“…Sample points of the evaluated BACs are also computed in this step. As explained before, sample points are calculated by solving the weighted sum model in Equation (9).…”

Section: Pareto Local Searchmentioning

confidence: 99%

“…The second phase of our approach is the same as in [2] which, in turn, makes use of (the improved) strategy introduced in [9]. In this phase, detailed in Algorithm 2, the associated MO-FMO is solved for each of the locally efficient BACs found in Phase 1.…”

Section: Second Phase: Exact Optimisation Of the Mo-fmo Problemmentioning

confidence: 99%

“…Therefore, it is simply not possible to approach this problem using enumeration strategies. Additionally, as reported in [9], state-of-the-art non-linear solvers, such as Knitro and Ipopt, can solve the BAO problem in a clinically acceptable time, with up to 12 available beams. Therefore, the LS algorithms proposed in this paper will only find a set ÂE ⊆ P N (K) that approximates the actual set of efficient BACs A E .…”

Section: Introductionmentioning

confidence: 98%

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Comparing Multi-Objective Local Search Algorithms for the Beam Angle Selection Problem

Cabrera‐Guerrero

Lagos

2022

Mathematics

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In intensity-modulated radiation therapy, treatment planners aim to irradiate the tumour according to a medical prescription while sparing surrounding organs at risk as much as possible. Although this problem is inherently a multi-objective optimisation (MO) problem, most of the models in the literature are single-objective ones. For this reason, a large number of single-objective algorithms have been proposed in the literature to solve such single-objective models rather than multi-objective ones. Further, a difficulty that one has to face when solving the MO version of the problem is that the algorithms take too long before converging to a set of (approximately) non-dominated points. In this paper, we propose and compare three different strategies, namely random PLS (rPLS), judgement-function-guided PLS (jPLS) and neighbour-first PLS (nPLS), to accelerate a previously proposed Pareto local search (PLS) algorithm to solve the beam angle selection problem in IMRT. A distinctive feature of these strategies when compared to the PLS algorithms in the literature is that they do not evaluate their entire neighbourhood before performing the dominance analysis. The rPLS algorithm randomly chooses the next non-dominated solution in the archive and it is used as a baseline for the other implemented algorithms. The jPLS algorithm first chooses the non-dominated solution in the archive that has the best objective function value. Finally, the nPLS algorithm first chooses the solutions that are within the neighbourhood of the current solution. All these strategies prevent us from evaluating a large set of BACs, without any major impairment in the obtained solutions’ quality. We apply our algorithms to a prostate case and compare the obtained results to those obtained by the PLS from the literature. The results show that algorithms proposed in this paper reach a similar performance than PLS and require fewer function evaluations.

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Section: Geud-based Mo-bao: Mathematical Formulationmentioning

confidence: 99%

“…Sample points of the evaluated BACs are also computed in this step. As explained before, sample points are calculated by solving the weighted sum model in Equation (9).…”

Section: Pareto Local Searchmentioning

confidence: 99%

Section: Second Phase: Exact Optimisation Of the Mo-fmo Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Comparing Multi-Objective Local Search Algorithms for the Beam Angle Selection Problem

Cabrera‐Guerrero

Lagos

2022

Mathematics

Self Cite

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“…• Algorithms based on scalarization which generates special scalarized MIPs that are then solved to optimality to obtain a new point and corresponding information about the Pareto frontier. The papers [2,18] discuss the convex case, while [3,4,19] focus on the linear case.…”

Section: Related Workmentioning

confidence: 99%

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Diessel

2021

Optimization

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Pareto frontiers of bicriteria continuous convex problems can be efficiently computed and optimal theoretical performance bounds have been established. In the case of bicriteria mixedinteger problems, the approximation of the Pareto frontier becomes, however, significantly harder. In this paper, we propose a new algorithm for approximating the Pareto frontier of bicriteria mixed-integer programs with convex constraints. Such Pareto frontiers are composed of patches of solutions with shared assignments for the discrete variables. By adaptively creating such a patchwork, our algorithm is able to create approximations that converge quickly to the true Pareto frontier. As a quality measure, we use the difference in hypervolume between the approximation and the true Pareto frontier. At least a certain number of patches is required to obtain an approximation with a given quality. This patch complexity gives a lower bound on the number of required computations. We show that our algorithm performs a number of optimization steps that are of a similar order as this lower bound. We provide an efficient MIP-based implementation of this algorithm. The efficiency of our algorithm is illustrated with numerical results showing that our algorithm has a strong theoretical performance guarantee while being competitive with other state-of-the-art approaches in practice.

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A test instance generator for multiobjective mixed-integer optimization

2023

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Application problems can often not be solved adequately by numerical algorithms as several difficulties might arise at the same time. When developing and improving algorithms which hopefully allow to handle those difficulties in the future, good test instances are required. These can then be used to detect the strengths and weaknesses of different algorithmic approaches. In this paper we present a generator for test instances to evaluate solvers for multiobjective mixed-integer linear and nonlinear optimization problems. Based on test instances for purely continuous and purely integer problems with known efficient solutions and known nondominated points, suitable multiobjective mixed-integer test instances can be generated. The special structure allows to construct instances scalable in the number of variables and objective functions. Moreover, it allows to control the resulting efficient and nondominated sets as well as the number of efficient integer assignments.

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

Cited by 10 publications

References 45 publications

Comparing Multi-Objective Local Search Algorithms for the Beam Angle Selection Problem

Comparing Multi-Objective Local Search Algorithms for the Beam Angle Selection Problem

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

A test instance generator for multiobjective mixed-integer optimization

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