Minimax Inequality Research Articles

Some new minimax inequalities for centered convex bodies

AbstractThe purpose of this manuscript is to derive some new minimax inequalities for centered convex bodies in $${\mathbb {R}}^{d}$$ R d . As consequences, new characterizations of ellipsoids will be established as well. Some open problems related to the Busemann-Petty list of problems will also be discussed.

Nash’s Existence Theorem for Non-Compact Strategy Sets

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions.

Necessary and sufficient conditions for the existence of fuzzy doubly strong equilibria in generalized fuzzy games

The concept of doubly strong equilibrium is a refinement of the concept of strong Nash equilibrium which represents the most desirable situations for generalized games, where all players get the best among the available outcomes. In this paper, we use necessary and sufficient conditions for the existence of solutions of the Ky Fan minimax inequality to establish necessary and sufficient conditions for the existence of fuzzy doubly strong equilibria in generalized fuzzy games. As consequences, we obtain sufficient conditions for the existence of (fuzzy) strong Nash equilibria in generalized (fuzzy) games, including generalized games with countable infinitely many players, which generalize existing existence theorems for strong Nash equilibria in games.

Variational inequalities, Ky Fan minimax inequality, and strong Nash equilibria in generalized games

Variational inequalities, Ky Fan minimax inequality, and strong Nash equilibria in generalized games

On the existence of solutions and optimal control for set-valued quasi-variational–hemivariational inequalities with applications

In this paper, we study the existence and optimal control of quasi-hemivariational inequalities by a method different from the one based on Minty's technique. We use an approach similar to the Galerkin method based on a minimax inequality formulation associated with the Brézis pseudomonotonicity notion of multi-valued operators, an implicit Browder–Tikhonov regularization method and a fixed point theorem. This leads us to avoid any kind of monotonicity-type conditions used in recent papers to obtain the convexity of the solution set of the variational selections. We provide applications to the optimal control of implicit obstacle problems of fractional Laplacian type involving a generalized gradient operator, and to the optimal control of contact problems for elastic locking materials. Our approach improves some recent results in the literature.

A minimax inequality for inscribed cones revisited

AbstractIn 1993, E. Lutwak established a minimax inequality for inscribed cones of origin symmetric convex bodies. In this work, we re-prove Lutwak’s result using a maxmin inequality for circumscribed cylinders. Furthermore, we explore connections between the maximum volume of inscribed double cones of a centered convex body and the minimum volume of circumscribed cylinders of its polar body.

Necessary and sufficient conditions for solutions of the Ky Fan minimax inequality and the non-emptiness of fuzzy cores in economies

In this paper, we provide a necessary and sufficient condition for the existence of solutions of the Ky Fan minimax inequality. We then apply this characterization to establish necessary and sufficient conditions for the non-emptiness of fuzzy cores and fuzzy α-cores in finite coalition production economies with infinite-dimensional commodity spaces. As consequences, we obtain existence theorems for cores and α-cores which generalize corresponding results in the literature, in particular, we derive the non-emptiness of fuzzy cores in finite coalition production economies with upper semicontinuous (possibly discontinuous) utility functions and compact feasible commodity allocation sets.

Minimax inequalities for functions with noncompact domain and diagonal GC-quasiconcavity and their applications

This paper introduces the notion of diagonal GC-quasiconcavity which generalizes the notions of quasiconcavity, CF-quasiconcavity, diagonal transfer quasiconcavity, C-quasiconcavity, diagonal C-concavity, and diagonal C-quasiconcavity. We first establish some theorems for the existence of ?-equilibrium of minimax inequalities for functions with noncompact domain and diagonal GC-quasiconcavity in topological spaces without linear structure. Next, we apply these results to characterize the existence of saddle points and solutions to the complementarity problem. Finally, we derive some intersection theorems and their equivalent forms.

Fuzzy Strong Nash Equilibria in Generalized Fuzzy Games with Application in Urban Public-Sports Services

In this paper, we apply the existence of solutions of the Ky Fan minimax inequality to establish the existence of fuzzy strong Nash equilibria in generalized fuzzy games and strong Nash equilibria in fuzzy coalition generalized games. We then show some applications of our existence results about the strong Nash equilibrium in urban public-sports services.

A general minimax inequality and its consequences

A general minimax inequality and its consequences

Existence of doubly strong equilibria in generalized games and quasi-Ky Fan minimax inequalities

Using new sufficient conditions for the existence of solutions to quasi-Ky Fan minimax inequalities in infinite dimensional setting, we prove the existence of strategy profiles, called doubly strong equilibria, which maximize players' preferences in generalized games. Our setting allows discontinuous and non-ordered preferences. The doubly strong equilibrium is a refinement of the strong equilibrium by Aumann. We show that our conditions do not imply the previous ones that guarantee the existence of strong equilibria.

Some Vector Ky Fan Minimax Inequalities with Nonconvex Domain

In this paper, by virtue of separation theorems of convex sets and scalarization functions, some minimax inequalities are first considered. As applications, some existence theorems of vector equilibrium problems with different order structures were also obtained.

A dynamical approach for the quantitative stability of parametric bilevel equilibrium problems and applications

In this paper, we primary establish Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems, DMEP for short, which we apply to obtain quantitative stability for parametric differential variational inclusions. Then, by involving key conditions on Fitzpatrick transform of equilibrium bifunctions introduced and studied in Chbani et al. [From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications. J Minimax Theory Appl. 2019;4:231–270], we derive further quantitative stability for a parametric bilevel equilibrium problem which is regarded in our approach as a limit problem of the dynamic model DMEP. The obtained abstract result on bilevel equilibria is thereby applied to parametric mathematical programs with equilibrium constraints. A specific applied model illustrating the meaning of the involved parameters is also discussed with respect to optimal control problems whose state equation is defined by a variational inequality. We also report a numerical illustration to highlight the convergence and estimation rates we obtained and support our theoretical results.

Collectively coincidence-type results and applications

Based on the well-known Brouwder fixed-point theorem in this paper we will present a variety of collectively coincidence-type results for general classes of maps. Our theory will automatically generate analytic alternatives and minimax inequalities.

A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces

An existence result for a generalized inequality over a possible unbounded domain in a finite-dimensional space is established. The proof technique allows to avoid any monotonicity assumption. We adapt a weak coercivity condition introduced in Castellani and Giuli (J Glob Optim 75:163–176, 2019) for a generalized game which extends an older one proposed by Konnov and Dyabilkin (J Glob Optim 49:575–577, 2011) for equilibrium problems. Our main result encompasses and generalizes several existence results for equilibrium, quasiequilibrium and fixed-point problems.

Coincidence Theory in the Coercive Case and Minimax Inequalities

We established coincidence results between maps with continuous selections and admissible maps. Both the compact and coercive cases were considered, and our argument relied on new coincidence ideas established recently by the author. Using our coincidence theory, we established new analytic alternatives, which then generate new minimax inequalities of the Neumann–Sion type.

Intersection theorems for generalized weak KKM set‐valued mappings with applications in optimization

AbstractIn this paper, we introduce the concept of generalized weak KKM mapping that is more general than many others encountered in the KKM theory. Then, two previous intersection theorems of the author are extended from weak KKM mappings to generalized weak KKM mappings. Applications of these results to set‐valued equilibrium problems and minimax inequalities are given in the last two sections.

Minimax inequalities for vector-valued functions in abstract convexity spaces

In this article, we first prove the extension of KKM lemma for generalized KKM mappings on abstract convexity spaces satisfying H 0-condition. Then, we give some generalized minimax inequalities for vector-valued functions by means of the generalized KKM theorem. Finally, we prove a cone-saddle point theorem as an application of our results.

Weak KKM set-valued mappings in hyperconvex metric spaces

In this paper, the concept of weak KKM set-valued mapping is extended from topological vector spaces to hyperconvex metric spaces. For these mappings we obtain several intersection theorems that prove to be useful in establishing existence criteria for weak and strong solutions of the general variational inequality problem and minimax inequalities.

Solvability and algorithms of generalized nonlinear variational-like inequalities in reflexive Banach spaces

This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.