In this article, we study linear complementarity problem with hidden Z-matrix. We extend the results of Fiedler and Ptak for the linear system in complementarity problem using game theoretic approa...

More on hidden Z-matrices and linear complementarity problem

The problem of incentive training of multi-agent systems in the game formulation for collective decision making under uncertainty is considered. Methods of incentive training do not require a mathematical model of the environment and enable decision making directly in the training process. Markov model of stochastic game is constructed and the criteria for its solution are formulated. An iterative Q-method for solving a stochastic game based on the numerical identification of a characteristic function of a dynamic system in space of state-action is described. Players’ current gains are determined by the method of randomization of payment Q-matrix elements. Mixed player strategies are calculated using the Boltzmann method. Pure strategies are determined on the basis of discrete random distributions given by mixed player strategies. The algorithm for stochastic game solving is developed and results of computer implementation of game Q-method are analyzed.

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Promoting training of multi agent systems.

We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts.

Variance and covariance of distributions on graphs

The existence of a linear order on a group that is compatible with the group structure generally requires transfinite methods. However, this can be circumvented by concentrating on the cons...

Ordering groups constructively

Solving binary-constrained mixed complementarity problems using continuous reformulations

In this paper, we revisit a result by Jurg et al. (Linear Algebra Appl 141:61–74, 1990) where the necessary and sufficient condition for a bimatrix game to be weakly completely mixed is given. We present an alternate proof of this result using linear complementarity approach. We extend this result to a generalization of bimatrix game introduced by Gowda and Sznajder (Int J Game Theory 25:1–12, 1996) via a generalization of linear complementarity problem introduced by Cottle and Dantzig (J Comb Theory 8:79–90, 1970). We further study completely mixed switching controller stochastic game (in which transition structure is a natural generalization of the single controller games) and extend the results obtained by Filar (Proc Am Math Soc 95:585–594, 1985) for completely mixed single controller stochastic game to completely mixed switching controller stochastic game. A numerical method is proposed to compute a completely mixed strategy for a switching controller stochastic game.

Completely Mixed Strategies for Generalized Bimatrix and Switching Controller Stochastic Game

Max-plus algebra has been applied to several fields like matrix algebra, cryptography, transportation, manufacturing, information technology and study of discrete event systems like subway traffic networks, parallel processing systems, telecommunication networks for many years. In this paper, we discuss various optimization problem using methods based on max-plus algebra, which has maximization and addition as its basic arithmetic operations. We present some sub-classes of mathematical optimization problems like linear programming, convex quadratic programming problem, fractional programming problem, bimatrix game problem and some classes of stochastic game problem in max algebraic framework and discuss various connections between max-plus algebra and optimization.

Max Plus Algebra, Optimization and Game Theory

In this paper, we introduce the notion of generalized positive subdefinite matrices of level k by generalizing the definition of generalized positive subdefinite matrices introduced by Crouzeix and Komlosi in [Applied optimization. Vol. 59, Dordrecht: Kluwer Academic Publications; 2001. p. 45–63]. The main motivation behind this generalization is to identify new subclasses of row sufficient matrices [Cottle, Pang and Venkateswaran, Linear Algebra Appl. 1989;114/115:231–249] which are not a subclass of copositive matrices. We establish new sufficient conditions for a matrix to be a non-copositive matrix and a non-copositive row sufficient matrix using the notion of generalized positive subdefinite matrices of level k. We present a new termination result for Lemke’s algorithm which supplements Jones’ result [Math Program. 1986;35:239–242].

On generalizations of positive subdefinite matrices and the linear complementarity problem

In this chapter, we study various mathematical programming problems in a common framework known as linear complementarity problem. Solving a linear complementarity problem depends on the properties of its underlying matrix class. In this chapter, we discuss matrix theoretic properties of some recent matrix classes encountered in linear complementarity literature and its processability using Lemke’s algorithm.

Dipti Dubey

Papers

Completely Mixed Strategies for Generalized Bimatrix and Switching Controller Stochastic Game

Max Plus Algebra, Optimization and Game Theory

On generalizations of positive subdefinite matrices and the linear complementarity problem

A Unified Framework for a Class of Mathematical Programming Problems