Some LCPs solvable in strongly polynomial time with Lemke’s algorithm

Abstract

We identify a class of Linear Complementarity Problems (LCPs) that are solvable in strongly polynomial time by Lemke's Algorithm (Scheme 1) or by the Parametric Principal Pivoting Method (PPPM). This algorithmic feature for the class of problems under consideration here is attributable to the proper selection of the covering vector in Scheme 1 or the parametric direction vector in the PPPM which leads to solutions of limited and monotonically increasing support size; such solutions are sparse. These and other LCPs may very well have multiple solutions, many of which are unattainable by either algorithm and thus are said to be elusive. The initial conditions imposed on the new matrix class identified in Sect. 2 are subsequently relaxed in later sections.

Some Strongly Polynomially Solvable Convex Quadratic Programs with Bounded Variables

“…The first such class of LCPs was discovered by Chandraskaran [2] for problems with a Zmatrix (i.e., with nonpositive off-diagonal elements). Subsequent extensions include [1,9]. Most recently, some quadratic sparse optimization problems with Stieltjes matrices (i.e., symmetric positive definite Z-matrices) and bounded variables are shown to be strongly polynomially solvable [5,6].…”