A Finite Algorithm for the Linear Exchange Model

It is shown that Lemke’s algorithm can be used to compute, in a finite number of steps, an equilibrium or reduction for the pure exchange model with linear utilities

A Finite Algorithm for the Linear Exchange Model

It is shown that Lemke's algorithm can be used to compute, in a finite number of steps, an equilibrium or reduction for the pure exchange model with linear utilities.

On Quadratic Programming

A procedure based on Lemke's algorithm is developed which either computes stationary points for general quadratic programs or else shows that the program has no optimum. If a general quadratic program has an optimum and satisfies a non-degeneracy condition then it is demonstrated that there are an odd number of stationary points.

The Linear Complementarity Problem

This study centers on the task of efficiently finding a solution of the linear complementarity problem: Ix - My = q, x \ge 0, y \ge 0, x \perp y. The main results are: (1) It is shown that Lemke's algorithm will solve (or show no solution exists) the problem for M \in L where L is a class of matrices, which properly includes (i) certain copositive matrices, (ii) certain matrices with nonnegative principal minors, (iii) matrices for bimatrix games. (2) If M \in L, if the system Ix - My = q, x \ge 0, y \ge 0 is feasible and nondegenerate, then the corresponding linear complementarity problem has an odd number of solutions. If M \in L and q > 0 then the solution is unique. (3) If for some M and every q \ge 0 the problem has a unique solution then M \in L and the problem has a solution for every q. (4) If M has nonnegative principal minors and if the linear complementarity with M and q has a nondegenerate complementary solution then the solution is unique. (5) If y TMy + y Tq is bounded below on y \ge 0 then the linear complementarity problem with M and q has a solution and Lemke's algorithm can be used to find such a solution. If, in addition, the problem is nondegenerate, then it has an odd number of solutions. (6) A procedure based on Lemke's algorithm is developed which either computes stationary points for general quadratic programs or else shows that the program has no optimum. (7) If a quadratic program has an optimum and satisfies a nondegeneracy condition then there are an odd number of stationary points.

A cellation of the Grassmann manifold

Motivated by the computation of equilibria in economic models with incomplete asset markets, a cellation of the Grassmann manifold is constructed by restricting a common atlas. The Grassmann manifold of m-planes in n-dimensional space is shown to be a union of n choose m congruent m(n - m)-dimensional topological disks whose interiors are disjoint.

Relationships of properties of piecewise affine maps over ordered fields

AbstractWith an emphasis on general ordered fields we survey relationships of properties of piecewise linear maps on vector spaces. The properties of interest are local and global variations of univalence, openness, onto condition, nonexpansiveness, and orientation and eigenvalue conditions on derivatives