S. K. Neogy

The generalized linear complementarity problem revisited

Given a vertical block matrixA, we consider in this paper the generalized linear complementarity problem VLCP(q, A) introduced by Cottle and Dantzig. We formulate this problem as a linear complementarity problem with a square matrixM, a formulation which is different from a similar formulation given earlier by Lemke. Our formulation helps in extending many well-known results in linear complementarity to the generalized linear complementarity problem. We also show that the class of vertical block matrices which Cottle and Dantzigs algorithm can process is the same as the class of equivalent square matrices which Lemkes algorithm can process. We also present some degree-theoretic results on a vertical block matrix.

Mathematical programming and game theory for decision making

Mathematical Programming and Its Applications in Finance Linear Programs with Totally Unimodular Coefficient Matrix Interior Point Method for Convex Quadratic Programming Analysis of Sets of Constraints, Traveling Salesman Problem and Tolerance-Based Algorithms Pedigree Polytope One-Defective Vertex Coloring Problem Complementarity Problem Fuzzy Twin Support Vector Machines for Pattern Classification Minimum Sum of Absolute Errors Regression Hedging Against the Market with No Short Selling Mathematical Programming and Electrical Network Analysis Dynamic Optimal Control Policy Forecasting for Supply Chain and Portfolio Management Variational Analysis in Bilevel Programming Game Engineering Games of Connectivity Robust Feedback Nash Equilibrium De Facto Delegation and Proposer Rules Bargaining Set in Effectivity Function Dynamic Oligopoly as a Mixed Large Game -- Toy Market, Balanced Games, Market Equilibrium for Combinatorial Auctions and the Matching Core of Non-negative TU Games Continuity, Manifolds and Arrows Social Choice Problem Mixture Class of Stochastic Games.

PIVOTING ALGORITHMS FOR SOME CLASSES OF STOCHASTIC GAMES: A SURVEY

In this paper, we survey the recent literature on computing the value vector and the associated optimal strategies of the players for special cases of zero-sum stochastic games, or in computing a Nash equilibrium point and the corresponding stationary strategies of the players for special cases of nonzero-sum stochastic games, using finite-step algorithms based on pivoting. Examples of finite-step pivoting algorithms are the various simplex-type algorithms, such as the primal simplex or dual simplex method for solving the linear programming problem or Lemkes or Lemke-Howsons algorithm for solving the linear complementarity problem. Also included are Lemke-type algorithms for solving various generalisations of the linear complementarity problem. The survey also includes a few new results and observations.

More on positive subdefinite matrices and the linear complementarity problem

Abstract In this article, we consider positive subdefinite matrices (PSBD) recently studied by J.-P. Crouzeix et al. [SIAM J. Matrix Anal. Appl. 22 (2000) 66] and show that linear complementarity problems with PSBD matrices of rank ⩾2 are processable by Lemkes algorithm and that a PSBD matrix of rank ⩾2 belongs to the class of sufficient matrices introduced by R.W. Cottle et al. [Linear Algebra Appl. 114/115 (1989) 231]. We also show that if a matrix A is a sum of a merely positive subdefinite copositive plus matrix and a copositive matrix, and a feasibility condition is satisfied, then Lemkes algorithm solves LCP( q , A ). This supplements the results of Jones and Evers.

On the classes of fully copositive and fully semimonotone matrices

Abstract In this paper we consider the class C 0 f of fully copositive and the class E 0 f of fully semimonotone matrices. We show that C 0 f matrices with positive diagonal entries are column sufficient. We settle a conjecture made by Murthy and Parthasarathy to the effect that a C 0 f ∩ Q 0 matrix is positive semidefinite by providing a counterexample. We finally consider E 0 f matrices introduced by Cottle and Stone (Math. Program. 27 (1983) 191–213) and partially address Stones conjecture to the effect that E 0 f ∩ Q 0 ⊆ P 0 by showing that E 0 f ∩ D c matrices are P 0 , where D c is the Doverspike class of matrices.

Vertical linear complementarity and discounted zero-sum stochastic games with ARAT structure

Abstract.In this paper we consider a two-person zero-sum discounted stochastic game with ARAT structure and formulate the problem of computing a pair of pure optimal stationary strategies and the corresponding value vector of such a game as a vertical linear complementarity problem. We show that Cottle-Dantzig’s algorithm (a generalization of Lemke’s algorithm) can solve this problem under a mild assumption.

Modeling, computation and optimization

Modeling a Jamming Game for Wireless Networks (A Garnaev) Existence of Nash Networks in the One-Way Flow Model of Network Formation (J Derks et al.) Strategic Advertisement with Externalities: A New Dynamic Approach (R Joosten) A New Axiomatization of the Shapley Value for TU-Games in Terms of Semi-Null Players Applied to 1-Concave Games (T S H Driessen) Coextrema Additive Operators (T Ui et al.) Models without Main Players (T S Arthanari) Weak Convergence of an Iterative Scheme with a Weaker Coefficient Condition (Y Kimura) Complementarity Modeling and Game Theory: A Survey (S K Neogy & A K Das) and other papers.

An Optimization Model to Determine Master Designs and Runs for Advertisement Printing

In this paper we consider a common optimization problem faced by a printing company while designing masters for advertisement material. A printing company may receive from various customers, advertisements for their products and services and their demand is for a specified number of copies to be printed. In a particular case, the printer receives these orders to be delivered next week from the customers, until the Thursday of a week. By Monday the printed copies have to be delivered to the customers. These advertisement items of the various customers are to be printed on large sheets of papers of specified standard sizes. The size is called a k-up if k items can be printed on one sheet. It is a given constraint that only items of the same size can be loaded on a master. This constraint results in a decomposition of the original problem of designing masters into many sub-problems, one for each size. The objective is to minimize the number of masters required while meeting the requirements of the customers. We formulate this optimization problem mathematically, discuss the computational issues and present some heuristic approaches for solving the problem.

Vertical Block Hidden Z -Matrices And The Generalized Linear Complementarity Problem

In this paper we introduce vertical block hidden Z-matrices and study their minimality and complementarity properties.

On discounted AR–AT semi-Markov games and its complementarity formulations

In this paper, we introduce a class of two-person finite discounted AR–AT (Additive Reward–Additive Transition) semi-Markov games (SMGs). We provide counterexamples to show that AR–AT and AR–AT–PT (Additive Reward–Additive Transition Probability and Time) SMGs do not satisfy the ordered field property. Some results on AR–AT–AITT (Additive Reward–Additive Transition and Action Independent Transition Time) and AR–AIT–ATT (Additive Reward–Action Independent Transition and Additive Transition Time) games are obtained in this paper. For the zero-sum games, we prove the ordered field property and the existence of pure stationary optimals for the players. Moreover, such games are formulated as a vertical linear complementarity problem (VLCP) and have been solved by Cottle-Dantzig’s algorithm under a mild assumption. We illustrate that the nonzero-sum case of such games do not necessarily have pure stationary equilibria. However, there exists a stationary equilibria which has at most two pure actions in each state for each player.