The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

Bazgan, Cristina; Ruzika, Stefan; Thielen, Clemens; Vanderpooten, Daniel

doi:10.1007/s00224-021-10066-5

Cited by 11 publications

(18 citation statements)

References 19 publications

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“…In this section, we investigate whether the results obtained in Section 3 can be transfered to the case of maximization. It is known that obtaining approximations using the weighted sum scalarization is more challenging for maximization problems than for minimization problems since the set of supported solutions does not yield any approximation guaranteein general [3]. We will see that this is also the case when using general ordering cones to obtain approximations.…”

Section: ⊓ ⊔ 4 Structural Results For Maximization Problemsmentioning

confidence: 84%

“…For biobjective optimization problems with convex feasible sets and linear objective functions, an efficient algorithm for computing (1 + ε)-approximations is studied in [4]. For an extensive study of the approximation quality achievable by the weighted sum scalarization for multiobjective minimization and maximization problems in general, see [3].…”

Section: Related Workmentioning

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: ⊓ ⊔ 4 Structural Results For Maximization Problemsmentioning

confidence: 84%

Section: Related Workmentioning

confidence: 99%

Approximating Biobjective Minimization Problems Using General Ordering Cones

Herzel¹,

Helfrich²,

Ruzika³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Thus, multi-objective optimization (MOO) needs special emphasis. However, many researchers have adopted formulations, like weighted sum approach [21,22] to transform an MOO problem into a SOO problem. This is an efficient way to reduce the complexity and computational intensiveness of an optimization problem.…”

Section: Introductionmentioning

confidence: 99%

A novel MOALO-MODA ensemble approach for multi-objective optimization of machining parameters for metal matrix composites

Kalita

Kumar

Chakraborty

2023

Multiscale and Multidiscip. Model. Exp. and Des.

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Machining of metal matrix composites (MMCs) has now become an essential task in shaping them to their final usable forms for various industrial applications. These machining operations may range from drilling of simple through holes to cutting and shaping the composites into complex shapes.Determination of the optimal process parameters during their machining presents a combinatorial optimization problem, which may turn out to be more complex due to involvement of various material dependent parameters of the MMCs, like particle size, reinforcement percentage etc. The importance of optimization increases manifold due to conflicting settings of different process parameters for attaining the desired quality characteristics. In this paper, two nature-inspired optimization algorithms, i.e. multi-objective antlion optimization (MOALO) and multi-objective dragonfly algorithm (MODA) are applied for multi-objective optimization of various process parameters during machining of MMCs. To mitigate uncertainties in global optimality of the predicted Pareto optimal fronts arising due to stochastic nature of the metaheuristics, an ensemble approach integrating MOALO and MODA techniques is proposed here. It is demonstrated with the help of two case studies that MOALO-MODA ensemble is far superior to its individual counterparts and thus, can be employed for development of robust and reliable Pareto optimal fronts.

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“…As a consequence, scalarization techniques are a key concept in multiobjective optimization: They often yield (weakly) efficient solutions and they are used as subroutines in algorithms for solving or approximating multiobjective optimization problems. Unsurprisingly, there exists a vast amount of research concerning both exact and approximate solutions methods that use scalarizations as building blocks, see Bökler and Mutzel (2015); Holzmann and Smith (2018); Klamroth et al (2015); Wierzbicki (1986) and Bazgan et al (2022); Daskalakis et al (2016); Diakonikolas and Yannakakis (2008); Glaßer et al (2010a,b); Halffmann et al (2017) and the references therein for a small selection.…”

Section: Introductionmentioning

confidence: 99%

An approximation algorithm for a general class of multi-parametric optimization problems

et al. 2022

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In a widely-studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. Then, the goal is to provide an optimal solution set, i.e., a set containing an optimal solution for each non-parametric problem obtained by fixing a parameter vector. For many multi-parametric optimization problems, however, an optimal solution set of minimum cardinality can contain super-polynomially many solutions. Consequently, no polynomial-time exact algorithms can exist for these problems even if $$\textsf {P}=\textsf {NP}$$ P = NP . We propose an approximation method that is applicable to a general class of multi-parametric optimization problems and outputs a set of solutions with cardinality polynomial in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric version and provides an approximation guarantee that is arbitrarily close to the approximation guarantee of the approximation algorithm for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, our algorithm is an FPTAS. Further, we show that, for any given approximation guarantee, the minimum cardinality of an approximation set is, in general, not $$\ell $$ ℓ -approximable for any natural number $$\ell $$ ℓ less or equal to the number of parameters, and we discuss applications of our results to classical multi-parametric combinatorial optimizations problems. In particular, we obtain an FPTAS for the multi-parametric minimum s-t-cut problem, an FPTAS for the multi-parametric knapsack problem, as well as an approximation algorithm for the multi-parametric maximization of independence systems problem.

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The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

Cited by 11 publications

References 19 publications

Approximating Biobjective Minimization Problems Using General Ordering Cones

Approximating Biobjective Minimization Problems Using General Ordering Cones

A novel MOALO-MODA ensemble approach for multi-objective optimization of machining parameters for metal matrix composites

An approximation algorithm for a general class of multi-parametric optimization problems

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