A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization

Bouza, Gemayqzel; Quintana, Ernest; Tammer, Christiane

doi:10.1007/s10013-019-00359-1

Cited by 17 publications

(8 citation statements)

References 38 publications

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“…The main difference is that, in Step 2, the authors use the well known Hiriart-Urruty functional and the support of a so called generator of the dual cone instead of ψ e , respectively. However, in in our framework, the functional ψ e is a particular case of those employed in the other methods, see [4,Corollary 2]. Thus, the equivalence in the case of vector optimization problems of the three algorithms is obtained.…”

Section: Algorithm 1 Descent Methods In Set Optimizationmentioning

confidence: 87%

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

Bouza,

Quintana,

Tammer

2021

Preprint

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In this paper, we study a first order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems, and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the PhD thesis by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms,

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Section: Algorithm 1 Descent Methods In Set Optimizationmentioning

confidence: 87%

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

Bouza,

Quintana,

Tammer

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…the functional ψ e is a particular case of those employed in the other methods, see [4,Corollary 2]. Thus, the equivalence in the case of vector optimization problems of the three algorithms is obtained.…”

Section: Algorithm 1 Descent Methods In Set Optimizationmentioning

confidence: 92%

“…Definition 3. 4 We say thatx is a stationary point of (SP ) if there exists a nonempty set Q ⊆ Px such that the following assertion holds:…”

Section: Theorem 31 Suppose Thatx Is a Local Weakly Minimal Solution Of (Sp ) Thenmentioning

confidence: 99%

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

Bouza

Quintana

Tammer

2021

J Optim Theory Appl

Self Cite

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In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020).

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“…Hence, we minimize here the well-known Tammer-Weidner-functional [11,14] which has many important properties, see for instance [17], and which turns out to be very useful in many theoretical and numerical approaches to multi-objective optimization. It is also the base of a very general scalarization in multiobjective optimization and covers several other scalarizations as special cases, see [5] and the recent review [1]. Let (t k+ , x k+ ) be a solution of (TW ).…”

Section: Basic Definitions and Algorithm Mhtmentioning

confidence: 99%

Representation of the Pareto front for heterogeneous multi-objective optimization

2019

J. Appl. Numer. Optim.

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Optimization problems with multiple objectives which are expensive, i. e., where function evaluations are time consuming, are difficult to solve. Finding at least one locally optimal solution is already a difficult task. In case only one of the objective functions is expensive while the others are cheap, for instance, analytically given, this can be used in the optimization procedure. Using a trust-region approach and the Tammer-Weidner-functional for finding descent directions, in [19] an algorithm was proposed which makes use of the heterogeneity of the objective functions. In this paper, we present three heuristic approaches, which allow to find additional optimal solutions of the multiobjective optimization problem and by that representations at least of parts of the Pareto front. We present the related theoretical results as well as numerical results on some test instances.

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A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization

Cited by 17 publications

References 38 publications

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

Representation of the Pareto front for heterogeneous multi-objective optimization

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