Covers and approximations in multiobjective optimization

Vanderpooten, Daniel; Weerasena, Lakmali; Wiecek, Margaret M.

doi:10.1007/s10898-016-0426-4

Cited by 9 publications

(8 citation statements)

References 54 publications

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“…More specifically, we investigate how optimal (or approximate) solutions with respect to general ordering cones can be used to achieve an approximation guarantee with respect to the usual Pareto cone. In contrast to the results by Vanderpooten et al [19] about approximation guarantees carrying over from smaller to larger ordering cones, we show that an approximation with respect to some fixed ordering cone containing the Pareto cone does not straightforwardly yield an approximation with respect to the Pareto cone (i.e., in the classical sense). We introduce the concept of γ-supportedness as a generalization of both supportedness and efficiency.…”

Section: Our Contributioncontrasting

confidence: 99%

“…However, the definition as stated in Definition 2.8 directly generalizes to arbitrary cones for more than two objectives while the alternative characterization is universally applicable only in the biobjective case. A very general definition of approximation in multiobjective optimization with respect to arbitrary cones and a further characterization of when the two mentioned definition approaches are equivalent are given by Vanderpooten et al [19]. They also present various results generalizing the following observation about approximations:…”

Section: Approximationmentioning

confidence: 98%

“…Vanderpooten et al [19] introduce a general framework modeling a variety of notions of approximation in the context of general ordering cones, including the concepts considered here. They provide conditions under which an approximation with respect to some cone is an approximation with respect to some other cone containing it.…”

Section: Related Workmentioning

confidence: 99%

“…Also, most of the definitions and observations stated in Section 2 easily transfer to the case of more than two objectives. For details, we refer to [19].…”

Section: Conclusion and Additional Notesmentioning

confidence: 99%

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Approximating Biobjective Minimization Problems Using General Ordering Cones

Herzel¹,

Helfrich²,

Ruzika³

et al. 2021

Preprint

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This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering α-approximations for all closed convex ordering cones of a fixed inner angle γ ∈ [ π 2 , π], an approximation guarantee between α and 2α is achieved, which depends continuously on γ. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation.Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.

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Section: Our Contributioncontrasting

confidence: 99%

Section: Approximationmentioning

confidence: 98%

Section: Related Workmentioning

confidence: 99%

“…Also, most of the definitions and observations stated in Section 2 easily transfer to the case of more than two objectives. For details, we refer to [19].…”

Section: Conclusion and Additional Notesmentioning

confidence: 99%

See 2 more Smart Citations

Approximating Biobjective Minimization Problems Using General Ordering Cones

Herzel¹,

Helfrich²,

Ruzika³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In this section, we consider solution quality metrics from the point of view of comparing algorithms on a test problem for which all function values and the true solution are known, and hence quantities such as g(Ê (t )) can be calculated exactly as a function of t. We formulate two solution quality metrics for the MOSO context: coverage error and hypervolume difference. Many other MOO metrics could be adapted for MOSO; see Audet et al (2018), Faulkenberg and Wiecek (2010), Jiang et al (2014), Vanderpooten et al (2017), Wu and Azarm (2001), and Zitzler et al (2003). Sayin (2000) and Eichfelder (2008, pp.…”

Section: Algorithmic Efficiency and Solution Quality Metricsmentioning

confidence: 99%

An Introduction to Multiobjective Simulation Optimization

Hunter

Applegate

Arora

et al. 2019

ACM Trans. Model. Comput. Simul.

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The multiobjective simulation optimization (MOSO) problem is a nonlinear multiobjective optimization problem in which multiple simultaneous and conflicting objective functions can only be observed with stochastic error. We provide an introduction to MOSO at the advanced tutorial level, aimed at researchers and practitioners who wish to begin working in this emerging area. Our focus is exclusively on MOSO methods that characterize the entire efficient or Pareto-optimal set as the solution to the MOSO problem; later, this set may be used as input to the broader multicriteria decision-making process. Our introduction to MOSO includes an overview of existing theory, methods, and provably convergent algorithms that explicitly control sampling error for (1) MOSO on finite sets, called multiobjective ranking and selection; (2) MOSO with integer-ordered decision variables; and (3) MOSO with continuous decision variables. In the context of integer-ordered and continuous decision variables, we focus on methods that provably converge to a local efficient set under the natural ordering. We also discuss key open questions that remain in this emerging field.

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Using scalarizations for the approximation of multiobjective optimization problems: towards a general theory

et al. 2023

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We study the approximation of general multiobjective optimization problems with the help of scalarizations. Existing results state that multiobjective minimization problems can be approximated well by norm-based scalarizations. However, for multiobjective maximization problems, only impossibility results are known so far. Countering this, we show that all multiobjective optimization problems can, in principle, be approximated equally well by scalarizations. In this context, we introduce a transformation theory for scalarizations that establishes the following: Suppose there exists a scalarization that yields an approximation of a certain quality for arbitrary instances of multiobjective optimization problems with a given decomposition specifying which objective functions are to be minimized/maximized. Then, for each other decomposition, our transformation yields another scalarization that yields the same approximation quality for arbitrary instances of problems with this other decomposition. In this sense, the existing results about the approximation via scalarizations for minimization problems carry over to any other objective decomposition—in particular, to maximization problems—when suitably adapting the employed scalarization. We further provide necessary and sufficient conditions on a scalarization such that its optimal solutions achieve a constant approximation quality. We give an upper bound on the best achievable approximation quality that applies to general scalarizations and is tight for the majority of norm-based scalarizations applied in the context of multiobjective optimization. As a consequence, none of these norm-based scalarizations can induce approximation sets for optimization problems with maximization objectives, which unifies and generalizes the existing impossibility results concerning the approximation of maximization problems.

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Covers and approximations in multiobjective optimization

Cited by 9 publications

References 54 publications

Approximating Biobjective Minimization Problems Using General Ordering Cones

Approximating Biobjective Minimization Problems Using General Ordering Cones

An Introduction to Multiobjective Simulation Optimization

Using scalarizations for the approximation of multiobjective optimization problems: towards a general theory

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