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

Bouza, Gemayqzel; Quintana, Ernest; Tammer, Christiane

doi:10.1007/s10957-021-01887-y

Cited by 10 publications

(4 citation statements)

References 32 publications

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“…Recently, robust vector optimization based on set orders is widely used in the uncertain optimization environment [23,24]. It is noteworthy that the robust vector equilibrium principles considered in this paper are all based on vector order.…”

Section: Discussionmentioning

confidence: 99%

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

2023

Axioms

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With the progress of society and the diversification of transportation modes, people are faced with more and more complicated travel choices, and thus, multi-criteria route choosing optimization problems have drawn increased attention in recent years. A number of multi-criteria traffic network equilibrium problems have been proposed, but most of them do not involve data uncertainty nor computational methods. This paper focuses on the methods for solving robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of the robust vector equilibrium and the robust vector equilibrium with respect to the worst case are introduced, respectively. For the robust vector equilibrium, an equivalent min–max optimization problem is constructed. A direct search algorithm, in which the step size without derivatives and redundant parameters, is proposed for solving this min–max problem. In addition, we construct a smoothing optimization problem based on a variant version of ReLU activation function to compute the robust weak vector equilibrium flows with respect to the worst case and then find robust vector equilibrium flows with respect to the worst case by using the heaviside step function. Finally, extensive numerical examples are given to illustrate the excellence of our algorithms compared with existing algorithms. It is shown that the proposed min–max algorithm may take less time to find the robust vector equilibrium flows and the smoothing method can more effectively generate a subset of the robust vector equilibrium with respect to the worst case.

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Section: Discussionmentioning

confidence: 99%

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

2023

Axioms

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show abstract

“…Recently, robust vector optimization based on set orders is widely used in the uncertain optimization environment [22,23]. It is noteworthy that the robust vector equilibrium principles considered in this paper are all based on vector order.…”

Section: Discussionmentioning

confidence: 99%

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

Xu¹,

Ma²

2023

Preprint

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This paper focuses on robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of robust vector equilibrium and robust vector equilibrium with respect to worst case are introduced, respectively. To generate a subset of robust vector equilibrium, an equivalent min-max optimization problem is constructed. In addition, a direct search algorithm, in which the step size without derivatives and redundant parameters, for solving this min-max problem is proposed. To find a subset of robust vector equilibrium with respect to worst case, an equivalent smoothing optimization problem based on a variant version of ReLU activation function is established. Numerical experiments are also presented to expound the advantages of the algorithms.

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“…The results are mostly of interest in the case in which the underlying spaces are Banach, but do not necessarily satisfy the Asplund condition. Using our results, it is possible to derive necessary optimality conditions for the solutions of set-valued optimization problems in the setting of general Banach spaces analogously to the procedure in [16] for Asplund spaces.…”

Section: Discussionmentioning

confidence: 99%

On Clarke’s subdifferential of marginal functions

Bouza

Quintana

Tammer

2021

J. Appl. Numer. Optim.

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

Cited by 10 publications

References 32 publications

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

On Clarke’s subdifferential of marginal functions

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