S. R. Mohan

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.

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.

Cone complementarity problems with finite solution sets

We introduce the notion of a complementary cone and a nondegenerate linear transformation and characterize the finiteness of the solution set of a linear complementarity problem over a closed convex cone in a finite dimensional real inner product space. In addition to the above, other geometrical properties of complementary cones have been explored.

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.

Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R 0-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.

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.

Linear Complementarity and the Irreducible Polystochastic Game with the Average Cost Criterion When One Player Controls Transitions

We consider the polystochastic game in which the transition probabilities depend on the actions of a single player and the criterion is the limiting average of the expected costs for each player. Using linear complementarity theory, we present a computational scheme for computing a set of stationary equilibrium strategies and the corresponding costs for this game with the additional assumption that under any choice of stationary strategies for the players the resulting one step transition probability matrix is irreducible. This work extends our previous work on the computation of a set of stationary equilibrium strategies and the corresponding costs for a polystochastic game in which the transition probabilities depend on the actions of a single player and the criterion is the total discounted expected cost for each player.

Algorithms For The Generalized Linear Complementarity Problem With A Vertical Block Z -Matrix

In this paper, we consider the generalized linear complementarity problem VLCP