A finite characterization of K -matrices in dimensions less than four

The class of real n × n matrices M , known as K -matrices, for which the linear complementarity problem w − Mz = q, w ≥ 0, z ≥ 0, w T z =0 has a solution whenever w − Mz =q, w ≥ 0, z ≥ 0 has a solution is characterized for dimensions n <4. The characterization is finite and ‘practical’. Several necessary conditions, sufficient conditions, and counterexamples pertaining to K -matrices are also given. A finite characterization of completely K -matrices ( K -matrices all of whose principal submatrices are also K -matrices) is proved for dimensions <4.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47913/1/10107_2005_Article_BF01589438.pd

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of order n for n ≥ 2—he has shown that to solve the problem of order n , either of the two methods goes through 2 n pivot steps before termination.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47905/1/10107_2005_Article_BF01588254.pd