Lemke paths are often used in the solution of nonlinear programming problems. We investigate a number of properties of Lemke paths, motivated by the d-step conjecture for linear programming. Some negative results are presented, including families of Lemke paths, for which the length of the shortest grows exponentially, joining pairs of vertices of a sequence of polytopes.

We use recent results on algorithms for Markov decision problems to show that a canonical form for a matrix with the generalized P-property can be computed, in some important cases, by a strongly polynomial algorithm.Keywords Markov decision problem · Polytope · Linear programming · P-matrix Mathematics Subject Classification 90C33 (90C05, 91A15)

We prove that convex geometries of convex dimension n that satisfy two properties satisfied by nondegenerate sets of points in the plane, may have no more than 2 n−1 points. We give examples of such convex geometries that have n 4 + n 2 + n 0 points.

The Linear Complementarity Problem can be shown to be equivalent to inverting a certain piecewise linear mapping. We give bounds on the degree of this mapping and present a class of matrices that generates maps of high degree.

We call a directed acyclic graph a CIdigraph if a certain affine semigroup ring defined by it is a complete intersection. We show that if D is a 2-connected CI-digraph with cycle space of dimension at least 2, then it can be decomposed into two subdigraphs, one of which can be taken to have only one cycle, that are CI-digraphs and are glued together on a directed path. If the arcs of the digraph are the covering relations of a poset, this is the converse of a theorem of Boussicault, Feray, Lascoux and Reiner [2]. The decomposition result shows that CI-digraphs can be easily recognized. 1991 AMS Mathematics subject classification. 14M10, 05C25, 05C20. Keywords and phrases. complete intersection, cycle basis, directed graph.