Stackelberg equilibria via variational inequalities and projections

Nagy, Szilárd

doi:10.1007/s10898-012-9971-7

Cited by 7 publications

(7 citation statements)

References 11 publications

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“…We have proven that the relationship between variational point and equilibrium points is obtained for geodesic convex payoff functions, extending the results obtained for convex payoff functions given by Nagy [20]. Let us illustrate this property with an example.…”

Section: Application: Stackelberg Equilibrium Problem Via Variationalsupporting

confidence: 74%

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Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

Garzón¹,

Osuna-Gómez²,

Rufián-Lizana³

et al. 2020

Preprint

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This article has two objectives. Firstly, we will use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of Vector Optimization Problem within the novel field of the Hadamard manifolds. Previously, we will introduce the concepts of generalized approximate geodesic convex functions and illustrate them with examples. We will see the minimum requirements under which critical points, solutions of Stampacchia and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we will show an economical application, using again solutions of the variational problems to identify with Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

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Section: Application: Stackelberg Equilibrium Problem Via Variationalsupporting

confidence: 74%

“…In 2013, Nagy [20] studied the existence of Stackelberg equilibria point using appropriate variational inequalities in Euclidean spaces.…”

Section: Introductionmentioning

confidence: 99%

Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

Garzón¹,

Osuna-Gómez²,

Rufián-Lizana³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Following Nagy [18], assume that f 1 , f 2 : R N × R N → R are the payoff/loss functions for two players, and K 1 , K 2 ⊂ R N are their strategy sets. It is well known that the framework of Stackelberg equilibrium problem can be modelled by the following bi-level mathematical programming problem:…”

Section: Existence Of Solutions For a Class Of Generalized Stackelbermentioning

confidence: 99%

“…Applying the variational inequality theory and the fixed point theorem, Nagy [18] studied the existence and location of Stackelberg equilibrium problem under the assumptions that f 1 and f 2 are both smooth functions. Moreover, Han and Huang [13] showed the existence of solutions for the Stackelberg equilibrium problem without the smoothness by employing the lower semicontinuity of the set-valued mapping R SE (x).…”

Section: Existence Of Solutions For a Class Of Generalized Stackelbermentioning

confidence: 99%

Lower semicontinuity of solution mappings for parametric fixed point problems with applications

Han

Huang

2017

Operations Research Letters

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In this paper, we establish the lower semicontinuity of the solution mapping and of the approximate solution mapping for parametric fixed point problems under some suitable conditions. As applications, the lower semicontinuity result applies to the parametric vector quasi-equilibrium problem, and allows to prove the existence of solutions for generalized Stackelberg games.The semicontinuity of solution mappings of vector equilibrium problems has been investigated by several authors, see [1-4, 6, 9, 12-15, 17] and the references therein. Recently, in order to show the semicontinuity of the solution mappings for the parametric (vector) quasi-equilibrium problems, all the solution mappings of the parametric fixed point problems are assumed to be lower semicontinuous in the literature [1][2][3]. We note that in the literature mentioned above, the authors have not given any conditions to guarantee the lower semicontinuity of the solution mappings of the parametric fixed point problems. On the other hand, it is difficult to obtain the explicit solutions for some real problems when the data concerned with the problems are perturbed by noise and so the mathematical models are usually solved by numerical methods for approximating the exact solutions. Therefore, one nature question is: can we provide conditions ensuring the lower semicontinuity of the (approximate) solution mappings?The main purpose of this paper is to make a new attempt to establish the lower semicontinuity of the solution mapping and of the approximate solution mapping for parametric fixed point problems under suitable conditions. The rest of the paper is organized as follows. Section 2 presents some necessary notations and lemmas. In Section 3, we establish the lower semicontinuity of the solution mapping and of the approximate solution mapping for parametric fixed point problems. In Section 4, the lower semicontinuity result applies to the parametric vector quasi-equilibrium problem, and allows to prove the existence of solutions for generalized Stackelberg games.

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“…More recently, in 2019, in Ruiz-Garzón et al [22], we studied the constrained vector optimization problem as a particular case of the equilibrium vector with constraints problem on Hadamard manifolds. In 2013, Nagy [23] studied the existence of Stackelberg equilibria points using appropriate variational inequalities in Euclidean spaces. In 2019, Wang et al [24] related the mixed variational inequality with the Nash equilibrium problem on Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

et al. 2020

View full text Add to dashboard Cite

This article has two objectives. Firstly, we use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of generalized approximate geodesic convex functions and illustrated them with examples. We see the minimum requirements under which critical points, solutions of Stampacchia, and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we show an economical application, again using solutions of the variational problems to identify Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

show abstract

Stackelberg equilibria via variational inequalities and projections

Cited by 7 publications

References 11 publications

Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

Lower semicontinuity of solution mappings for parametric fixed point problems with applications

Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

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