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John Cotrina

Publications and source records attributed to John Cotrina.

18 recordsLinked to original sources
arXiv2026

Sequential Stability of the Value Function and the Solution Mapping in Berge's Maximum Theorem via Variational Convergence

Berge's maximum theorem ensures the continuity of the value function and the upper semicontinuity of the solution mapping in parametric optimization problems. This theorem plays a central role in optimization theory, game theory, and dynamic programming. Motivated by the inherent inaccuracies in optimization data, this paper investigates the stability of such problems under sequential perturbations of both the objective function and the feasible mapping. The analysis focuses on the convergence of sequences of value functions and solution mappings via variational approximations of the data. To this end, we employ lower and upper continuous, epi- and hypo-convergence notions for functions, together with lower and upper continuous and graphical convergence notions for multifunctions. In addition, we study some relationships among these types of convergence and provide examples and counterexamples associated with the corresponding notions. Our results extend and complement existing stability results in the literature. We provide applications to generalized Nash equilibrium problems, where stability is obtained via a direct approach, as well as to finite-horizon dynamic programming models under novel perturbation assumptions.

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arXiv2025

Lipschitz continuity of expected value under decision-dependent uncertainty with moving support

This paper addresses the problem of stochastic optimization with decision-dependent uncertainty, a class of problems where the probability distribution of the uncertain parameters is influenced by the decision-maker's actions. While recent literature primarily focuses on solving or analyzing these problems by directly imposing hypotheses on the distribution mapping, we explore in this work some of these properties for a specific construction by means of the moving support and a density function. The construction is motivated by the Bayesian approach to bilevel programming, where the response of a follower is modeled as the uncertainty, drawn from the moving set of optimal responses, which depends on the leader's decision. Our main contribution is to establish sufficient conditions for the Lipschitz continuity of the expected value function. We show that Lipschitz continuity can be achieved when the moving support is a Lipschitz continuous set-valued map with full-dimensional, convex, compact values, or when it is the solution set of a fully linear parametric problem. We also provide an example showing that the sole Lipschitz assumption on the moving set itself is not sufficient and that additional conditions are necessary.

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arXiv2023

The generalized Nash game proposed by Rosen

We deal with the generalized Nash game proposed by Rosen, which is a game with strategy sets that are coupled across players through a shared constraint. A reduction to a classical game is shown, and as a consequence, Rosen's result can be deduced from the one given by Arrow and Debreu. We also establish necessary and sufficient conditions for a point to be a generalized Nash equilibrium employing the variational inequality approach. Finally, some existence results are given in the non-compact case under coerciveness conditions.

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arXiv2023

Generalized Ordinal Nash Games: Variational Approach

It is known that the generalized Nash equilibrium problem can be reformulated as a quasivariational inequality. Our aim in this work is to introduce a variational approach to study the existence of solutions for generalized ordinal Nash games, that is, generalized games where the player preferences are binary relations that do not necessarily admit a utility representation.

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arXiv2023

Existence and uniqueness of maximal elements for preference relations: Variational approach

In this work, we reformulate the problem of existence of maximal elements for preference relations as a variational inequality problem in the sense of Stampacchia. Similarly, we establish the uniqueness of maximal elements using a variational inequality problem in the sense of Minty. In both of these approaches, we use the normal cone operator to find existence and uniqueness results, under mild assumptions. In addition, we provide an algorithm for finding such maximal elements, which is inspired by the steepest descent method for minimization. Under certain conditions, we prove that the sequence generated by this algorithm converges to a maximal element.

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arXiv2022

Inverse maximum theorems and some consequences

We deal with inverse maximum theorems, which are inspired by the ones given by Aoyama, Komiya, Li et al., Park and Komiya, and Yamauchi. As a consequence of our results, we state and prove an inverse maximum Nash theorem and show that any generalized Nash game can be reduced to a classical Nash game, under suitable assumptions. Additionally, we show that a result by Arrow and Debreu, on the existence of solutions for generalized Nash games, is actually equivalent to the one given by Debreu-Fan-Glicksberg for classical Nash games, which in turn is equivalent to Kakutani-Fan-Glisckberg's fixed point theorem.

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arXiv2022

A note on coupled constraint Nash games

In this note we are interested in a relevant generalized Nash equilibrium problem, which was proposed by Rosen in 1965. An existence result is established in the general setting of quasiconvexity, which is independent from the one given by Aussel and Dutta in 2008.

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arXiv2022

Remarks on Pseudo-continuity

In this work we extend a maximum theorem proposed by Morgan and Scalzo. We also show some results of minimax inequalities which are equivalent to the famous Ky Fan minimax inequality. Additionally, we prove that the existence result of Nash equilibria proposed by Morgan and Scalzo is actually equivalent to a classical result in the literature.

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arXiv2020

On the construction of maximal $p$-cyclically monotone operators

In this paper we deal with the construction of explicit examples of maximal $p$-cyclically monotone operators. To date, there is only one instance of an explicit example of a maximal 2-cyclically monotone operator that is not maximal monotone. We present a systematic way to construct this kind of examples, along with several explicit examples.

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arXiv2019

Remarks on $p$-monotone operators

In this paper, we deal with three aspects of $p$-monotone operators. First we study $p$-monotone operators with a unique maximal extension (called pre-maximal), and with convex graph. We then deal with linear operators, and provide characterizations of $p$-monotonicity and maximal $p$-monotonicity. Finally we show that the Brezis-Browder theorem preserves $p$-monotonicity in reflexive Banach spaces.

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arXiv2019

Coerciveness condition for quasi-equilibrium problems

A quasi-equilibrium problem is an equilibrium problem where the constraint set does depend on the reference point. It generalizes important problems such as quasi-variational inequalities and generalized Nash equilibrium problems. We study the existence of equilibria on unbounded sets under a coerciveness condition adapted from one specific for quasi-variational inequalities recently proposed by Aussel and Sultana. We discuss the relation of our results with others that are present in the literature.

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arXiv2019

Quasi-equilibrium problems with generalized monotonicity

In this work, we propose a new existence result for quasi-equilibrium problems using generalized monotonicity in an infinite dimensional space. Also, we show that the notions of generalized monotonicity can be characterized in terms of solution sets of equilibrium problems and convex feasibility problems. Moreover, we show that the concept of pseudomonotonicity and upper sign property are related under suitable assumptions.

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arXiv2019

Quasi-Equilibrium Problems with Non-self Constraint Map

In 2016 Aussel, Sultana and Vetrivel developed the concept of projected solution for quasi-variational inequality problems and projected Nash equilibrium. We introduce a new concept of solution for quasi-equilibrium problems and we study the existence of such solutions. Additionally, as a consequence of our results, we give existence results of projected solutions for quasi-optimization problems, quasi-variational inequalities problems and generalized Nash equilibrium problems.

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arXiv2017

A note on quasi-equilibrium problems

The purpose of this paper is to prove the existence of solutions of quasi-equilibrium problems without any generalized monotonicity assumption. Additionally, we give an application to quasi-optimization problems.

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arXiv2017

Time-Dependent Generalized Nash Equilibrium Problem

We prove an existence result for the time-dependent generalized Nash equilibrium problem under generalized convexity using a fixed point theorem. Furthermore, an application to the dynamic abstract economy is considered.

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arXiv2016

The Pseudomonotone Polar for Multivalued Operators

In this work, we study the pseudomonotonicity of multivalued operators from the point of view of polarity, in an analogous way as the well-known monotone polar due to Martínez-Legaz and Svaiter, and the quasimonotone polar recently introduced by Bueno and Cotrina. We show that this new polar, adapted for pseudomonotonicity, possesses analogous properties to the monotone and quasimonotone polar, among which are a characterization of pseudomonotonicity, maximality and pre-maximality. Furthermore, we characterize the notion of $D$-maximal pseudomonotonicity introduced by Hadjisavvas. We conclude this work studying the connections between pseudomonotonicity and variational inequality problems.

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arXiv2016

On maximality of quasimonotone operators

We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martinez-Legaz and Svaiter. We first recover several properties similar to the monotone polar, including a characterization in terms of normal cones. Next, we use it to analyze certain aspects of maximal (in the sense of graph inclusion) quasimonotonicity, and its relation to the notion of maximal quasimonotonicity introduced by Aussel and Eberhard. Furthermore, we study the connections between quasimonotonicity and Minty Variational Inequality Problems.

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