Minimax Research Articles

The Complexity of Pure Maxmin Strategies in Two-Player Extensive-Form Games

Extensive-form games model strategic interaction between players, with an emphasis on the sequential aspect of decision-making: players take turns to move until an ending is reached, and receive a reward according to which ending is reached. We study the complexity of computing the pure maxmin value for such games, i.e. the maximum reward that a player can guarantee by playing a pure strategy, whatever their opponents play. We focus on two-player and two-team games and perform a systematic study depending on the degree of imperfect information of each player or team: perfect information, perfect recall, or perfect recall for each agent in a team (which we call multi-agent perfect recall). For each combination, we settle the complexity of deciding whether the maxmin value is at least as high as a given threshold. We give a complete complexity picture for three orthogonal settings: games represented explicitly by their game tree; games represented compactly by game rules, for which we propose two new formalisms; games in which the set of strategies of the opponents is restricted to a known set of opponent models.

Python Development of Algorithm to find Solution in Pure Strategies of Antagonistic Matrix Game

The article provides an algorithm for finding the optimal solution to the matrix antagonistic game in pure strategies. The algorithm implements the search for the upper and lower prices of the matrix game by implementing the maximal and minimax strategies of the players. The algorithm is encoded using the functionality of the Python programming language. The algorithm is tested using a numerical example.

Distributionally Robust Appointment Scheduling That Can Deal With Independent Service Times

Consider a single server that should serve a given number of customers during a fixed period. The appointment scheduling problem (ASP) determines the schedule of planned appointments that minimizes some cost function that accounts for both the cost of idle times and the cost of waiting. When service time distributions are fully specified, the ASP presents a much-investigated computationally challenging stochastic program. When service time distributions are only partially specified, one can apply distributionally robust optimization (DRO) to find the schedule that minimizes costs in worst-case circumstances. We assume that only the mean, mean absolute deviation and range of the service times are known and develop a DRO method that finds the optimal (mini–max) schedule. For independent service times, the min–max problem becomes nonlinear and difficult, if not impossible, to solve exactly. Existing DRO methods for ASP with partial information (such as mean and variance), therefore, consider relaxations that allow correlations between service times. Such relaxations have major repercussions, as the worst-case scenario will then be highly correlated. Our method thus deals with independent service times and finds a robust schedule as the solution to a linear program. We identify several new structural features of optimal robust schedules. We also apply the method to model extensions including sequencing and alternative objective functions.

High-precision minmax solution of the two-center Dirac equation

High-precision minmax solution of the two-center Dirac equation

Development of an AI-Powered Chess Engine Using Minimax Algorithm and Genetic Algorithm for Evaluation Function

Development of an AI-Powered Chess Engine Using Minimax Algorithm and Genetic Algorithm for Evaluation Function

Spectral analysis of eigenvalue problem with the (p(x),q(x))-Laplacian

In this study, a spectral problem that involves the Robin (a(x),b(x))-Laplacian on a bounded domain in Rn is analyzed. With the help of the Ljusternik-Schnirelmann principle, it is established that an increasing sequence of eigenvalues exists. Using the min-max principle, we also established that the infimum of the spectrum is zero, and all the eigenvalues are non-negative. It was further proved that there exists a principal eigenvalue for this problem and that the spectrum is open.

Min-max theory for capillary surfaces

Abstract We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature 𝑐, and with smooth boundary contacting at any given constant angle 𝜃. Moreover, if 𝑐 is nonzero and 𝜃 is not π 2 \frac{\pi}{2} , then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.

Stochastic Resource Allocation Problems: Minmax and Maxmin Solutions

Problem definition: Min-max or max-min objective criteria naturally arise in many resource allocation applications, especially when the attendant goal involves distribution of a limited resource across multiple subsystems, locations, or activities. Each has a different consumption rate of the resource. In this setting, the decision maker is interested in optimizing the system performance, which can be measured in terms of the smallest or largest order statistic across subsystems. Examples include the allocation of health resources to maximize the first runout time or to minimize the maximum delay. Methodology/results: In the literature, there are no unified theoretical and computational results on the optimal allocations for these problems. Thus, in this paper, we develop key theoretical properties of the attendant stochastic resource allocation problems, providing a technical condition for the objective functions to have favorable properties, upon which we derive novel optimal allocation rules with clear interpretation. Managerial implications: The proportional allocation rule can be interpreted as a robust solution when the full distribution of the interarrival time is unknown. It generally has poor performance, especially when the resource consumption rates vary across locations, which happens regularly in practice. Our allocation rule is exact and should be used for the allocation of scarce resources—that is, when the number of resources per location is limited, which is often encountered in healthcare settings.

APPLICATION OF A NEURONETWORK FOR DETERMINING THE TYPE OF ELEMENTS OF A SYMMETRICAL COMPENSATION DEVICE OF AN UNSYMMETRICAL SYSTEM WITH A ZERO WIRE

In article the possibility of using neural networks in the field of increasing the energy performance of a four-wire power supply system with an uneven load in the phases is being investigated. An uneven load in the phases causes asymmetry of currents in the network and contributes to the increase in the current in the neutral wire, which has an extremely negative effect on both the supply itself and its consumers. To eliminate asymmetry and reduce the current in the neutral wire, you can connect a symmetrical compensating device. Such a device is a set of reactive elements, the parameters of which are determined by search optimization. To determine the type of the required element, the defined parameters are recalculated. That is, the solution of such a problem consists of two component sub-problems – structural and parametric synthesis. This approach provides a high accuracy of calculations, but has a significant drawback: the calculations are cumbersome. In order to simplify the solution of the synthesis problem, it makes sense to determine the type of elements using neural networks, which will significantly reduce the time and resources spent on calculating the values of the parameters of the symmetrical compensating device. The subject of the article is the study of the possibility of using neural networks to predict the types of reactive elements of the symmetrical compensating device. During of the study, the parameters and type of neural network were determined, which provide the most accurate prediction of the topology of the structure of the symmetric-compensating device. The input parameters of the neural network were formed from sets consisting of eight parameters – resistances and inductances of the transmission lines and the neutral wire. The target matrix was formed from a set of data sets consisting of six elements containing information on the types of elements connected (0 – capacitor, 1 – inductance) between phases and between phases and the neutral wire. During research, the results of a quasi-solution were obtained, the values of which turned out to be commensurate with the accurate calculations for determining the structure of the symmetrical compensating device of the power supply system with a zero wire. This indicates the high quality of the developed neural network. Applying the minimax strategy to the received results provides an opportunity to reduce the received values to 0 and 1 to ensure clarity of the results obtained by the neural network.

A Multiobjective Optimization Algorithm for Fluid Catalytic Cracking Process with Constraints and Dynamic Environments

The optimization problems in a fluid catalytic cracking process with dynamic constraints and conflicting objectives are challenging due to the complicated constraints and dynamic environments. The decision variables need to be reoptimized to obtain the best objectives when dynamic environments arise. To solve these problems, we established a mathematical model and proposed a dynamic constrained multiobjective optimization evolution algorithm for the fluid catalytic cracking process. In this algorithm, we design an offspring generation strategy based on minimax solutions, which can explore more feasible regions and converge quickly. Additionally, a dynamic response strategy based on population feasibility is proposed to improve the feasible and infeasible solutions by different perturbations, respectively. To verify the effectiveness of the algorithm, we test the algorithm on ten instances based on the hypervolume metric. Experimental results show that the proposed algorithm is highly competitive with several state-of-the-art competitors.

Unjust inequalities. Is maximin the answer?

Abstract A maximin criterion of distributive justice provides an attractive way of articulating a concern for equality and a concern for efficiency. It recommends that one should equalize income, expected advantages, opportunities or whatever else matters for the sake of distributive justice up to the point from which further equalization would make the worse off even worse off once its consequences are taken into account. But is the maximin criterion egalitarian enough? Does it pay enough attention to the impact of inequalities on political power, to the equal dignity of all citizens or to the value of fraternity? Is the maximin criterion not instead too egalitarian? Does it pay enough attention to a priority legitimately claimed by one’s own community, to democratically expressed preferences or to the value of reciprocity? This contribution clarifies the maximin concept, discusses these various objections and concludes that a maximin approach remains essential to identify, if not just inequalities, at least justifiable ones.

Efficient Aperture Fill Time Correction for Wideband Sparse Array Using Improved Variable Fractional Delay Filters.

To solve the problem of aperture fill time (AFT) for wideband sparse arrays, variable fractional delay (VFD) FIR filters are applied to eliminate linear coupling between spatial and time domains. However, the large dimensions of the filter coefficient matrix result in high system complexity. To alleviate the computational burden of solving VFD filter coefficients, a novel multi-regultion minimax (MRMM) model utilizing the sparse representation technique has been presented. The error function is constrained by the introduction of L2-norm and L1-norm regularizations within the minimax criterion. The L2-norm effectively resolves the problems of overfitting and non-unique solutions that arise in the sparse optimization of traditional minimax (MM) models. Meanwhile, the use of multiple L1-norms enables the optimal design of the smallest sub-filter number and order of the VFD filter. To solve the established nonconvex model, an improved sequential-alternating direction method of multipliers (S-ADMM) algorithm for filter coefficients is proposed, which utilizes sequential alternation to iteratively update multiple soft-thresholding problems. The experimental results show that the optimized VFD filter reduces system complexity significantly and corrects AFT effectively in a wideband sparse array.

Fairness in accessibility of public service facilities

Our paper studies a fair stochastic facility location problem with congestion under the max-min principle. We analyze the price of fairness in a two-location problem and develop a tractable optimization framework for the general multi-location setting. Evaluating the fair solution against the utilitarian solution on Buffalo's demographic data reveals that implementing a fair solution can substantially improve fairness measures (up to 98%) with relatively limited impact on the overall service quality.

Mathematical models for determining the optimum cyber security strategy of intelligent computer networks of railway electrical supply distances

A study of the problem of the security of information resources of intelligent computer networks for power supply management in railway power generation was conducted. The logical structure of an intelligent computer network is presented, reflecting the topological characteristics of the power supply distance, in the form of a graph. A conceptual approach to the organization of an optimal cyber security strategy is proposed. On the based of Pukhov's differential transformations, differential mathematical models are proposed for determining, in analytical form, the state probabilities of graph nodes. Based on the minimax principle, methods have been developed to determine the optimal cyber security strategy, which allows to achieve the specified security indicators. A criterion for ensuring information security is formulated, which allows determining the state of cyber security of each graph node and the probability of this node being in this state.

Nontrivial solutions for resonance quasilinear elliptic systems

Abstract We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically ( p , q ) \left(p,q) -linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its C 2 {C}^{2} regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.

Game Playing in Artifical Intelligence

Abstract: This paper focuses on the how Game Playing is an important domain of Artificial intelligence .AI is also used in game playing. Various researchers have extensively contributed for computer game –playing methods. Games don’t require much knowledge: the only knowledge we need to provide is the rules, legal moves and the condition of winning or losing Game . Both players try to win the game . so both of them try to make the best possible move at each turn. Shannon’s paper on chess-playing and Samuel’s checkers are considered as land marks in the area of computer game playing. We also explain in this paper major components of game playing program , static evaluation function generator, game playing strategies: i) minimax strategy ii) minimax strategy with alpha – beta cutoffs.

New neighborhoods in robustness and least favorable pairs for special capacities

We propose a new neighborhood of distributions for robust inference, which is generated by a certain special capacity with three constants. It covers not only the familiar neighborhoods defined in terms of ε-contamination, total variation and their combination, but also their intersections. It is shown that the neighborhood is also obtained from contaminating some neighborhood which consists of absolutely continuous distributions. We derive the explicit forms of Radon-Nikodym derivatives and the least favorable pairs for such special capacities and neighborhoods on the real line, which generalize the earlier results of Huber and Rieder for minimax testing, and reinforce the general results of Bednarski.

A regularization strategy for discontinuity when modelling coupled water and heat flow in freezing unsaturated soil

Freezing tightly couples the water and heat flow. In most porous media, the interface between liquid and frozen water is not sharp and a slushy zone is present. There are two distinct approaches for mathematical modelling of freezing. It is the equilibrium approach which allows an instant freezing under given conditions and non-equilibrium approach where a specific timing in the freezing process is considered. In this contribution, we specifically target the equilibrium approach.The key mathematical model for the equilibrium approach is the Clausius–Clapeyron equation, which allows the derivation of a soil freezing curve relating temperature to pressure head. Implementing freezing soil accurately is not a straight-forward. Using the Clausius–Clapeyron equation creates a discontinuity in the freezing rate and latent heat at the freezing point. Little attention has been paid to the adequate description of the numerical treatment of this phenomenon and to the computational challenges that it poses. Numerical approximation of this discontinuous system is prone to spurious oscillations. In this contribution, we show the application of regularization of the discontinuous term. This treatment successfully stabilizes the computation and can remove oscillations. To avoid over-regularization, we present here a minimax strategy to determine optimal regularization parameters. We further compare an over-regularized setup with the non-equilibrium approach, where we show that a successful regularization is equivalent to incorporating a minimal timing to the freezing process. Finally – for validating our implementation and computational approach – experimental laboratory data of the volumetric water content and temperature profiles from previously published soil freezing experiments were represented here with the optimally regulized equilibrium approach.

Focus on informative graphs! Semi-supervised active learning for graph-level classification

Graph-level classification is a critical problem in social analysis and bioinformatics. Since annotated labels are typically costly, we intend to study this challenging task in semi-supervised scenarios with limited budgets. Inspired by the fact that active learning is capable of interactively querying an oracle to annotate a small number of informative examples in the unlabeled dataset, we develop a novel Semi-supervised active learning framework termed GraphSpa for graph-level classification. To make the most of labeling budgets, we propose an effective unlabeled data selection strategy that takes both local similarity and global semantic structure into account. Specifically, we first construct an adaptive queue with labeled samples and select informative samples that have a low degree of similarity to the queue using the Min-Max principle from the local view. Further, we introduce class prototypes and select samples with a large predictive loss discrepancy from the global view. To harness the full potential of unlabeled data, we develop a semi-supervised active learning framework on the basis of our fusion selection strategy coupled with graph contrastive learning during active learning. The effectiveness of our GraphSpa is validated against state-of-the-art methods through experimental results on diverse real-world benchmark datasets.

Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer

We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector \kappa=(\kappa_{1},\kappa_{2}) which characterises the domain at the edges. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases \kappa_{1}\ge0 and \kappa_{2}\in[-\kappa_{1},\kappa_{1}] . We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to \kappa . In particular, we show that for a given \kappa_{1}>0 , there is some h(\kappa_{1})>0 such that discrete spectrum exists for \kappa_{2}\in[-\kappa_{1},0)\cup(h(\kappa_{1}),\kappa_{1}] whereas it is empty for \kappa_{2}\in[0,h(\kappa_{1})] . The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.