Approximating multiobjective optimization problems: How exact can you be?

Cristina Bazgan (), Arne Herzel (), Stefan Ruzika (), Clemens Thielen () and Daniel Vanderpooten ()
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Cristina Bazgan: Université Paris-Dauphine, PSL Reseach University
Arne Herzel: University of Kaiserslautern-Landau
Stefan Ruzika: University of Kaiserslautern-Landau
Clemens Thielen: Weihenstephan-Triesdorf University of Applied Sciences
Daniel Vanderpooten: Université Paris-Dauphine, PSL Reseach University

Mathematical Methods of Operations Research, 2024, vol. 100, issue 1, No 2, 5-25

Abstract: Abstract It is well known that, under very weak assumptions, multiobjective optimization problems admit $$(1+\varepsilon ,\dots ,1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) -approximation sets (also called $$\varepsilon $$ ε -Pareto sets) of polynomial cardinality (in the size of the instance and in $$\frac{1}{\varepsilon }$$ 1 ε ). While an approximation guarantee of $$1+\varepsilon $$ 1 + ε for any $$\varepsilon >0$$ ε > 0 is the best one can expect for singleobjective problems (apart from solving the problem to optimality), even better approximation guarantees than $$(1+\varepsilon ,\dots ,1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) can be considered in the multiobjective case since the approximation might be exact in some of the objectives. Hence, in this paper, we consider partially exact approximation sets that require to approximate each feasible solution exactly, i.e., with an approximation guarantee of 1, in some of the objectives while still obtaining a guarantee of $$1+\varepsilon $$ 1 + ε in all others. We characterize the types of polynomial-cardinality, partially exact approximation sets that are guaranteed to exist for general multiobjective optimization problems. Moreover, we study minimum-cardinality partially exact approximation sets concerning (weak) efficiency of the contained solutions and relate their cardinalities to the minimum cardinality of a $$(1+\varepsilon ,\dots ,1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) -approximation set.

Keywords: Multiobjective optimization; Approximation; Efficient set; Intractability (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00186-023-00836-x

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