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53 pages, 1295 KiB

Open AccessReview

Selective Reviews of Bandit Problems in AI via a Statistical View

by Pengjie Zhou, Haoyu Wei and Huiming Zhang

Mathematics 2025, 13(4), 665; https://doi.org/10.3390/math13040665 - 18 Feb 2025

Cited by 3 | Viewed by 802

Reinforcement Learning (RL) is a widely researched area in artificial intelligence that focuses on teaching agents decision-making through interactions with their environment. A key subset includes multi-armed bandit (MAB) and stochastic continuum-armed bandit (SCAB) problems, which model sequential decision-making under uncertainty. This review [...] Read more.

Reinforcement Learning (RL) is a widely researched area in artificial intelligence that focuses on teaching agents decision-making through interactions with their environment. A key subset includes multi-armed bandit (MAB) and stochastic continuum-armed bandit (SCAB) problems, which model sequential decision-making under uncertainty. This review outlines the foundational models and assumptions of bandit problems, explores non-asymptotic theoretical tools like concentration inequalities and minimax regret bounds, and compares frequentist and Bayesian algorithms for managing exploration–exploitation trade-offs. Additionally, we explore K-armed contextual bandits and SCAB, focusing on their methodologies and regret analyses. We also examine the connections between SCAB problems and functional data analysis. Finally, we highlight recent advances and ongoing challenges in the field. Full article

(This article belongs to the Special Issue Advances in Statistical AI and Causal Inference)

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10 pages, 223 KiB

Open AccessArticle

Nash’s Existence Theorem for Non-Compact Strategy Sets

by Xinyu Zhang, Chunyan Yang, Renjie Han and Shiqing Zhang

Mathematics 2024, 12(13), 2017; https://doi.org/10.3390/math12132017 - 28 Jun 2024

Viewed by 883

Abstract

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, [...] Read more.

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. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

25 pages, 10343 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Jordan-Type Inequalities and Stratification

by Miloš Mićović and Branko Malešević

Axioms 2024, 13(4), 262; https://doi.org/10.3390/axioms13040262 - 14 Apr 2024

Cited by 3 | Viewed by 2142

Abstract

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding [...] Read more.

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications, 2nd Edition)

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12 pages, 290 KiB

Open AccessArticle

A Discrete Characterization of the Solvability of Equilibrium Problems and Its Application to Game Theory

by Maria Isabel Berenguer, Domingo Gámez, Ana Isabel Garralda-Guillem and Manuel Ruiz Galán

Axioms 2023, 12(7), 666; https://doi.org/10.3390/axioms12070666 - 5 Jul 2023

Viewed by 1277

Abstract

We state a characterization of the existence of equilibrium in terms of certain finite subsets under compactness and transfer upper semicontinuity conditions. In order to derive some consequences on game theory—Nash equilibrium and minimax inequalities—we introduce a weak convexity concept. Full article

(This article belongs to the Special Issue Advances in Logic and Game Theory)

10 pages, 271 KiB

Open AccessArticle

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

by Tieying Huang and Jiuqiang Liu

Mathematics 2022, 10(20), 3784; https://doi.org/10.3390/math10203784 - 13 Oct 2022

Cited by 6 | Viewed by 1575

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

29 pages, 940 KiB

Open AccessArticle

Inertial Modification Using Self-Adaptive Subgradient Extragradient Techniques for Equilibrium Programming Applied to Variational Inequalities and Fixed-Point Problems

by Habib ur Rehman, Wiyada Kumam and Kamonrat Sombut

Mathematics 2022, 10(10), 1751; https://doi.org/10.3390/math10101751 - 20 May 2022

Cited by 10 | Viewed by 1853

Abstract

Equilibrium problems are articulated in a variety of mathematical computing applications, including minimax and numerical programming, saddle-point problems, fixed-point problems, and variational inequalities. In this paper, we introduce improved iterative techniques for evaluating the numerical solution of an equilibrium problem in a Hilbert [...] Read more.

Equilibrium problems are articulated in a variety of mathematical computing applications, including minimax and numerical programming, saddle-point problems, fixed-point problems, and variational inequalities. In this paper, we introduce improved iterative techniques for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. These techniques are based on two computing steps of a proximal-like mapping with inertial terms. We investigated two simplified stepsize rules that do not require a line search, allowing the technique to be carried out more successfully without knowledge of the Lipschitz-type constant of the cost bifunction. Once control parameter constraints are put in place, the iterative sequences converge on a particular solution to the problem. We prove strong convergence theorems without knowing the Lipschitz-type bifunction constants. A sequence of numerical tests was performed, and the results confirmed the correctness and speedy convergence of the new techniques over the traditional ones. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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11 pages, 289 KiB

Open AccessArticle

Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

by Ruofeng Rao

Mathematics 2021, 9(16), 1943; https://doi.org/10.3390/math9161943 - 14 Aug 2021

Cited by 4 | Viewed by 1840

Abstract

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin–Ayala competition model with impulsive disturbance. The conclusion obtained in this paper [...] Read more.

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin–Ayala competition model with impulsive disturbance. The conclusion obtained in this paper reduces the conservatism of the algorithm compared with the known literature, for the impulsive disturbance is not limited to impulsive control. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering)

9 pages, 235 KiB

Open AccessArticle

Coincidence Theory in the Coercive Case and Minimax Inequalities

by Donal O’Regan

Symmetry 2021, 13(7), 1220; https://doi.org/10.3390/sym13071220 - 7 Jul 2021

Viewed by 1635

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Mathematics)

16 pages, 312 KiB

Open AccessArticle

Strong Convergence of Extragradient-Type Method to Solve Pseudomonotone Variational Inequalities Problems

by Nopparat Wairojjana, Nuttapol Pakkaranang, Habib ur Rehman, Nattawut Pholasa and Tiwabhorn Khanpanuk

Axioms 2020, 9(4), 115; https://doi.org/10.3390/axioms9040115 - 13 Oct 2020

Cited by 7 | Viewed by 3047

Abstract

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, [...] Read more.

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)

24 pages, 1384 KiB

Open AccessArticle

A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

by Nopparat Wairojjana, Habib ur Rehman, Manuel De la Sen and Nuttapol Pakkaranang

Axioms 2020, 9(3), 101; https://doi.org/10.3390/axioms9030101 - 31 Aug 2020

Cited by 11 | Viewed by 3416

Abstract

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, [...] Read more.

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)

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