ON FULLY SEMIMONOTONE MATRICES

The class of fully semimonotone matrices is well known in the study of the linear complementarity problem. Stone [(1981) Ph.D. thesis, Dept. of Operations Research, Stanford University, Stanford, CA] introduced this class and conjectured that the principal minors of any fully semimonotone Q0-matrix are non-negative. While the problem is still open, Murthy and Parthasarathy [(1998) Math. Program. 82, 401–411] introduced the concept of incidence using which they proved that the principal minors of any matrix in the class of fully copositive Q0-matrices, a subclass of fully semimonotone Q0-matrices, are non-negative. In this paper, we study some properties of fully semimonotone matrices in connection with incidence. The main result of the paper shows that Stone's conjecture is true in the special case where the complementary cones have no partial incidence. We also present an interesting characterization of Q0 for matrices with a special structure. This result is very useful in checking whether a given matrix belongs to Q0 provided it has the special structure. Several examples are discussed in connection with incidence and Q0 property.