Approximating biobjective minimization problems using general ordering cones

Arne Herzel (), Stephan Helfrich (), Stefan Ruzika () and Clemens Thielen ()
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Arne Herzel: RPTU Kaiserslautern-Landau
Stephan Helfrich: RPTU Kaiserslautern-Landau
Stefan Ruzika: RPTU Kaiserslautern-Landau
Clemens Thielen: Weihenstephan-Triesdorf University of Applied Sciences

Journal of Global Optimization, 2023, vol. 86, issue 2, No 5, 393-415

Abstract: Abstract 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 $$\alpha $$ α -approximations for all closed convex ordering cones of a fixed inner angle $$\gamma \in \left[ \frac{\pi }{2}, \pi \right] $$ γ ∈ π 2 , π , an approximation guarantee between $$\alpha $$ α and $$2 \alpha $$ 2 α is achieved, which depends continuously on $$\gamma $$ 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 in general by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.

Keywords: Multiobjective optimization; Approximate Pareto set; Ordering cone; Supported solution; Efficient solution; 90C29; 68W25 (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-023-01276-x

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