Minimax Research Articles

This research proposes a method based on the greatest and shortest eigen fuzzy sets of fuzzy relations to evaluate the effectiveness of policies and strategies implemented in urban settlements aimed at increasing the livability and well-being of citizens. This assessment is performed by extracting population census data collected at the beginning and end of the analyzed period and grouping them by subzone, that is, for each of the urban areas into which the urban settlement is divided. The greatest eigen fuzzy set (GEFS) and the smallest eigen fuzzy set (SEFS) are determined symmetrically as min-max and max-min solutions of fuzzy relations; they are calculated to estimate the average impact of urban strategies on generating symptoms of growth in citizen well-being during the investigated period. The method is implemented on a GIS (Geographic Information System) platform and was tested to assess the effectiveness of local policies applied between 2011 and 2021 on the growth of employment rates and educational attainment in the city of Naples (Italy), partitioned by neighborhood. Our model, unlike other fuzzy-based models for evaluating the effectiveness of actions and strategies to improve the quality of life in urban settlements, does not allow for subjective interpretations based on the knowledge or experience of different stakeholders, but relies solely on measurements over time of characteristics collected during census activities. Furthermore, it is integrated into a GIS-based platform, providing additional capabilities for identifying the urban areas where the impact of local strategies and policies has been most significant and those most critical. The test results show that the proposed framework can be a valuable tool for supporting decision makers in evaluating the effectiveness of local actions and policies aimed at improving the livability and well-being of citizens.

Multi-objective process optimization is critical in intelligent manufacturing, especially where complex, nonlinear interactions among parameters can significantly affect product quality. This study demonstrates a data-driven approach to optimizing Refill Friction Stir Spot Welding (RFSSW) parameters for an AA2024-T3 aluminum alloy. First, a 3³ full-factorial design of experiments was conducted to generate training and validation data on three pivotal process variables: rotational speed, plunge depth, and welding time. Statistical analysis using ANOVA highlighted plunge depth as the most influential factor, alongside notable interaction effects among the parameters. To build predictive models of joint load capacity, six machine learning techniques (MLP, RBF, GPR, k-NN, SVR, and XGBoost) were evaluated via cross-validation. XGBoost delivered the most accurate predictions, reaching R² values up to 0.89 with the lowest MAE and RMSE. Model interpretation methods such as feature importance and SHAP confirmed the dominant role of plunge depth, as suggested by ANOVA. The crux of the study lies in a multi-objective optimization framework using the NSGA-II evolutionary algorithm, targeting maximum weld strength in two distinct shear-testing variants (pure shear and free shear). The procedure generated a Pareto frontier of optimal parameter sets, from which a maximin strategy selected a high-performing compromise solution. These findings underscore the value of combining statistical methods, advanced machine learning, and evolutionary optimization in refining solid-state joining processes. More broadly, this integrated methodology provides a robust template for intelligent manufacturing applications that require balancing multiple performance objectives under complex process conditions.

Minimax principles for lower semicontinuous functionals and applications to logarithmic Schrödinger System

Covariance and the min-max principle in relativistic quantum chemistry: minimal basis Dirac-Fock calculations on atoms up to Z = 18

Infrared-visible image fusion methods aim at generating fused images with good visual quality and also facilitate the performance of high-level tasks. Indeed, existing semantic-driven methods have considered semantic information injection for downstream applications. However, none of them investigates the potential for reciprocal promotion between pixel-wise image fusion and cross-modal feature fusion perception tasks from a macroscopic task-level perspective. To address this limitation, we propose a unified network for image fusion and semantic segmentation. MAFS is a parallel structure, containing a fusion sub-network and a segmentation sub-network. On the one hand, We devise a heterogeneous feature fusion strategy to enhance semantic-aware capabilities for image fusion. On the other hand, by cascading the fusion sub-network and a segmentation backbone, segmentation-related knowledge is transferred to promote feature-level fusion-based segmentation. Within the framework, we design a novel multi-stage Transformer decoder to aggregate fine-grained multi-scale fused features efficiently. Additionally, a dynamic factor based on the max-min fairness allocation principle is introduced to generate adaptive weights of two tasks and guarantee smooth training in a multi-task manner. Extensive experiments demonstrate that our approach achieves competitive results compared with state-of-the-art methods.

Abstract In this paper, we consider the normalized ground state solutions for the following biharmonic Choquard type problem \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u-\beta\Delta u=\lambda u+(I_\mu*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \\ \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^4),\\ \end{array} \right. \end{split} \end{align*} where $\beta\geq0$ , c > 0, $\lambda\in \mathbb{R}$ , $I_\mu=\frac{1}{|x|^\mu}$ with $\mu\in (0,4)$ , F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth in the sense of the Adams inequality. By using a minimax principle based on the homotopy stable family, we obtain that the above problem admits at least one normalized ground state solution.

To address the challenges of insufficient robustness in single-modal features and interference from cross-modal disparities in pedestrian re-identification under complex scenarios, we propose a novel network model that integrates joint attention mechanisms and multimodal features. Built upon a residual network backbone, the model introduces a cross-modal self-attention module to adaptively weight features from RGB, thermal infrared, and depth modalities. A multimodal feature fusion module is designed with three branches: intra-modal enhancement, cross-modal correlation, and modal discrepancy suppression, which together construct comprehensive pedestrian feature representations. During optimization, we introduce a combination of modal cosine cross-entropy loss, cross-modal triplet loss, center alignment loss, and modal consistency loss, updating the network using a min-max strategy. The proposed method achieves top-1 accuracy rates of 94.3% and 88.7% on the RegDB and SYSU-MM01 datasets, respectively, demonstrating its effectiveness in multimodal pedestrian re-identification scenarios.

The paper develops a theory of M-estimators optimizing the weighted L2-norm of the influence function. The specified criterion of the estimation quality is quite general and, in addition, allows obtaining solutions related to the class of redescending estimators, i.e., possessing the property of stability to asymmetric contamination. Such estimators, in particular, were studied within the framework of the locally stable approach of A.M. Shurygin, based on the analysis of the estimator instability functional (L2-norm of the influence function), or his approach based on the model of a series of samples with random point contamination (point Bayesian contamination model). In this paper, a compromise family of estimators is studied for which the optimized functional is a convex linear combination of two basic criteria. The compromise family is similar to the conditionally optimal family of estimators proposed by A.M. Shurygin, but the criteria used can be squares of the weighted L2-norms of the influence function with arbitrary pre-known weight functions. The considered subject area has remained little-studied to date. In the course of the research, we used a theory we had developed earlier, which describes the properties of estimators that optimize the weighted L2-norm of the influence function. As a result of the study, a number of properties of compromise estimators were obtained, and the uniqueness of the family elements was shown. A family member that delivers equal values to the criteria was considered separately: it was shown that this estimator corresponds to the saddle point of the optimized functional, and is also a minimax solution with respect to the basic criteria on the set of all regular score functions. The constructed theory is illustrated using the example of the problem of mathematical expectation estimating of a normal distribution under conditions of targeted malicious influence on a data set (similar to a data poisoning attack in malicious machine learning).

Due to incomplete and unattainable information, the data of inputs and outputs in production often cannot be obtained crisply but are represented by fuzzy data. Therefore, the data envelopment analysis (DEA) approach with precise data to fixed cost allocation is not applicable anymore, and the fuzzy DEA method is necessitated. This paper proposes the fuzzy DEA model based on the fuzzy expected values approach for the first measurement of fixed cost allocation, where the fuzzy inputs and fuzzy outputs are respectively weighted. It proves that there exist feasible allocation schemes that can render each DMU and the collection of all DMUs efficient. Additionally, with the help of the fuzzy expected values approach, the Max-min satisfaction degree principle is utilized to achieve the optimal solution to the fixed cost allocation in the fuzzy scenario. The proposed fuzzy expected values approach based on fuzzy DEA and satisfaction degree for the fixed cost allocation is illustrated by two numerical examples. It shows that the fuzzy DEA approach with the fuzzy expected values can crisply evaluate DMUs and avoid the comparison dilemma of fuzzy efficiencies, as well as determine a precise allocation plan of fixed costs that can make all DMUs DEA efficient.

The evolution of the mass use of artificial intelligence is analyzed, and the direction of research related to the optimization of the architecture and state modes of AI networks is substantiated. It is shown that the performance of AI networks largely depends on their topological characteristics. Their representation in the form of an unweighted and undirected graph is proposed, and a number of mathematical models for topology optimization are developed. The application of the minimax strategy for optimizing the modes of operation of AI networks in the case of the worst combination of the intensity of the transition action flow with an arbitrary counteraction flow law is substantiated. Based on the theory of differential transformations, differential mathematical models are proposed for optimizing the dynamics of the probabilities of the states of AI systems in time. To ensure the necessary productivity of information exchange in AI systems and take into account the conditional balance of traffic volume and sufficient bandwidth at all levels of the hierarchy, a new approach is proposed, which is based on methods for reducing topological network distances as the basis for reducing transit traffic

This paper explores the application of game theory in chess, examining its evolution from classical minimax algorithms to modern artificial intelligence approaches. The study investigates how game-theoretic principles have shaped strategic thinking in chess, analyzing key concepts like Nash equilibrium, minimax, and reinforcement learning. It also discusses the intersection of psychology and strategy, highlighting how human behavior complicates the theoretical model of perfect information. The paper concludes by reflecting on the role of AI in transforming chess and its implications for broader decision-making frameworks.

To improve the trajectory tracking performance of unmanned surface vehicle (USV), this article investigates the USV optimal control problem with the consideration of actuator wear. In the proposed algorithm, the USV control system is divide into kinematic subsystem and kinetic subsystem. In particular, corresponding performance indexes that looking forward to be optimized are defined for each subsystem. The related value functions, Hamilton-Jacobi-Bellman equations and optimal control policies are approximated by actor-critic neural networks. To reduce the wear of propeller and rudder, the event-triggered problem is considered as a zero-sum game solving problem, where the best control inputs and worst thresholds are delivered via minmax strategy. Also, the nonlinear uncertainties of the USV are approximated and environment disturbances are compensated in the value functions for better control performance. The USV closed-loop control system is proved semi-globally uniformly ultimately bounded stability via Lyapunov theory. Finally, a simulation case and harbor experiment are illustrated to verify the superiorities and engineering application values of the proposed algorithm.

In computerized adaptive testing (CAT), information-based item selection rules (ISRs), such as maximum Fisher information (MFI), often excessively rely on discriminating items, leading to unbalanced utilization of the item pool. To address this challenge, the present study introduced the MaxiMin Information (MMI) criterion, which is grounded in decision theory. MMI calculates each item's minimum information (Imin) within the current confidence interval (CI) of the trait level, selecting the item with the maximum Imin to be administered. For examinees with broader CIs (less precise trait estimates), MMI leans toward administering less discriminating items, which tend to yield larger Imin. Conversely, for narrower CIs, MMI aligns more closely with MFI by favoring items with higher discrimination. This indicates that MMI's item selection is tailored to each examinee based on his or her provisional trait estimate and its estimation precision. Five simulation studies were conducted to assess MMI's performance in CAT under various conditions. Results demonstrate that although MMI is comparable with other ISRs in terms of trait estimation precision, it excels in balancing item pool utilization. By fine-tuning confidence levels, MMI not only efficiently schedules the use of discriminating items toward the test's later stages to enhance test efficiency but also effectively adapts to different testing scenarios. From these findings, we generally recommend applying MMI with a confidence level of 95% to optimize item pool utilization without compromising trait estimation accuracy. With its evident advantages, MMI holds promise for practical applications, especially for high-stakes tests requiring utmost test efficiency and security.

The object of this research is the factors that significantly influence the competitiveness of enterprises operating in the modern high-tech society. The paper examines the business environment that actively uses modern mobile communication technologies. The relevance of this research stems from societal concerns associated with modern mobile communication technologies (3G, 4G, 5G) and the rapid development of 6G, which may pose potential risks. These risks can impact businesses that rely on such technologies in their operations. This paper proposes an approach to determining the dependence of business competitiveness on mobile communication technologies based on game theory. Performance matrices were constructed, and risk analysis was carried out according to the criteria of Wald, Savage, and Hurwitz. Potential operational strategies were analyzed in the context of environmental states, considering responses to market fluctuations and unpredictable factors. The influence of specific factors on enterprise competitiveness was assessed under conditions of complete uncertainty. To compare the impact of mobile communication technologies, a simulation model in C# was developed. The study considered 240 enterprises in the market of the Republic of Kazakhstan. Two scenarios were compared: the use of 4G versus 5G technology. The results were visualized as a model ranking enterprise based on the impact of mobile communication technology. A distinctive feature of the study is the identification of environmental states, which served as a basis for grouping risk factors by their influence on competitive position. The minimax and maximin principles were applied to describe enterprise behavior in a competitive environment. The simulation model was split up. The simulation model revealed skewed gains and shortcomings in the competitiveness of enterprises that were monitored. The proposed approach can be applied to business growth projects, marketing strategy enhancement, and automation of tasks aimed at improving competitiveness in enterprises across all forms of ownership. It is also applicable to banking and credit institutions in the justification and optimization of lending policies.

Stochastic vibration control of uncertain structures under random loading is an important problem and its minimax optimal control strategy remains to be developed. In this paper, a stochastic optimal control strategy for uncertain structural systems under random excitations is proposed, based on the minimax stochastic dynamical programming principle and the Bayes optimal estimation method with the combination of stochastic dynamics and Bayes inference. The general description of the stochastic optimal control problem is presented including optimal parameter estimation and optimal state control. For the estimation, the posterior probability density conditional on observation states is expressed using the likelihood function conditional on system parameters according to Bayes’ theorem. The likelihood is replaced by the geometrically averaged likelihood, and the posterior is converted into its logarithmic expression to avoid numerical singularity. The expressions of state statistics are derived based on stochastic dynamics. The statistics are further transformed into those conditional on observation states based on optimal state estimation. Then, the obtained posterior will be more reliable and accurate, and the optimal estimation will greatly reduce uncertain parameter domains. For the control, the minimax strategy is designed by minimizing the performance index for the worst-parameter system, which is obtained by maximizing the performance index based on game theory. The dynamical programming equation for the uncertain system is derived according to the minimax stochastic dynamical programming principle. The worst parameters are determined by the maximization of the equation, and the optimal control is determined by the minimization of the resulting equation. The minimax optimal control by combining the Bayes optimal estimation and minimax stochastic dynamical programming will be more effective and robust. Finally, numerical results for a five-story frame structure under random excitations show the control effectiveness of the proposed strategy.

We consider correlated equilibria in an adversarial environment, where an adversary can compromise the public signal used by the players for choosing their strategies, while players aim at detecting a potential attack as soon as possible to avoid loss of utility. We model the interaction between the adversary and the players as a zero-sum game and we derive the maxmin strategies for both the defender and the attacker using the framework of quickest change detection. We define a class of adversarial strategies that achieve the optimal trade-off between the impact and the detectability of the attack for the adversary and show that a generalized CUSUM scheme is asymptotically optimal for their detection. Our numerical results on the Sioux-Falls benchmark traffic routing game show that the proposed detection scheme can effectively limit the utility loss by a potential adversary.

Abstract In this article, we investigate the following non-linear Schrödinger (NLS) equation with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\ \displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega \end{cases} \end{equation*} coupled with a constraint condition: \begin{equation*} \int_{\Omega}|u|^2 dx=c, \end{equation*} where $\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, ν represents the unit outer normal vector to $\partial \Omega$ , c is a positive constant, and λ acts as a Lagrange multiplier. When the non-linearity f exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various non-linearities, including cases such as $f(u)=|u|^{p-2}u$ , $|u|^{q-2}u+ |u|^{p-2}u$ , and $-|u|^{q-2}u+|u|^{p-2}u$ , where $2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$ . The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.

In this article, we focus on solving the problem of online multiplayer differential games (MDGs) of Markov jump systems (MJSs) using a reinforcement learning (RL) method. We consider MDGs of MJSs from the following two scenarios. In the first scenario, we propose a distributed minmax strategy, where each player can derive their optimal control policy from distributed game algebraic Riccati equations (DGAREs) without prior knowledge of the policies adopted by other players, distinguishing it from existing RL algorithms. We design a novel online distributed RL algorithm to approximate the solution of DGAREs without completely knowing system dynamics and initial admissible control policy. The second scenario involves applying Nash strategy to address MDGs of MJSs. Different from existing synchronous RL algorithm, we propose a novel online asynchronous RL algorithm that employs asynchronous iterative calculations for both policy evaluation and policy improvement, incorporating the latest information into the iterative process. The convergence of the designed RL algorithms is rigorously analyzed. Finally, two inverted pendulum system applications validate the effectiveness of the proposed methods.

In complex markets, limited information makes it difficult for investors to accurately estimate true distribution of the risky asset return, which reducing model robustness. This paper focuses on the consumption investment problem under risky asset return distributional uncertainty. By integrating distributional uncertainty with Bayesian methods, the paper constructs a robust model. Using the Minimax principle and dynamic programming, the paper derives explicit solution strategies under the CARA utility function by solving a minimax optimization problem. Empirical analysis verifies the rationality and effectiveness of the proposed model.

Over the past three decades, the intensity-modulated radiotherapy (IMRT) has become a standard technique, enabling highly conformal dose distributions tailored to specific clinical objectives. Despite these advancements, IMRT treatment plans are significantly susceptible to uncertainties during both the planning and delivery phases. The most commonly used strategy to address these uncertainties is the margin-based or planning target volume (PTV) approach, which relies on the so-called dose cloud approximation. However, the PTV concept has notable limitations, particularly in complex scenarios where target volumes are superficial or located near critical structures. In contrast, the advent of intensity-modulated particle therapy has driven the development of robust optimization models, which have emerged as a promising alternative for managing uncertainties. Among these, the worst-case scenario or minimax strategy is the most widely employed. While minimax can be directly applied to photon treatments, its use in IMRT often leads to overly conservative plans or plans that are very similar to those obtained using the conventional margin-based PTVapproach. In this work, we present a robust optimization model particularly suitable for photon treatments. The new approach, called Cheap-Minimax, is a generalization of the minimax strategy used for particle therapy and aims to improve the balance between plan robustness and the price of robustness in terms of dose to organs at risk (OARs), an issue particularly pronounced in photontreatments. The c-minimax model was implemented in the MatRad treatment planning system, developed at the German Cancer Research Center (DKFZ). It was applied to 20 clinical cases, comprising 5 prostate cancer cases and 15 breast cancer cases. The results were compared with those obtained using the conventional minimax model and the PTV-basedapproach. For prostate cancer cases, the c-minimax model maintained a robustness comparable to the PTV approach, while achieving a 20% reduction in for the rectum and a 10% reduction in for the bladder compared to the minimax model. In breast cancer cases, the c-minimax model improved robustness by 23.7% relative to the PTV approach and by 18.2% compared to the minimax model. Additionally, the c-minimax model reduced for the ipsilateral lung by 3.7% and the mean heart dose by 1.2 Gy (20%) compared to minimax. Both the c-minimax and minimax models reduced skin dose by 10.9 Gy (18.9%) and 11.1 Gy (19.3%), respectively, compared to the PTVapproach. The c-minimax model successfully overcomes the limitations of the PTV approach and the over-conservativeness of the minimax model, demonstrating significant advantages in managing uncertainties in complex cases, such as breast cancer. By providing superior robustness compared to PTV and reducing OAR doses relative to minimax, the model offers a flexible and clinically feasible strategy to enhance treatment quality. The marked reduction in high-dose regions (hotspots) in superficial tissues and skin highlights its potential to lower toxicity risks and improve patient outcomes. These results provide quantitative evidence of the practical benefits of robustness-compromise-oriented approaches inIMRT.