On semimonotone star matrices and linear complementarity problem

In this article, we introduce the class of semimonotone star ($E_0^s$) matrices. We establish importance $E_0^s$-matrices in context complementarity theory. show that principal pivot transform $E_0^s$-matrix is not necessarily $E_0^s$ general. However, prove $\tilde{E_0^s}$-matrices, a subclass with some additional conditions, $E_0^f$ by showing $P_0.$ LCP$(q, A)$ can be processable Lemke's algorithm if $A\in \tilde{E_0^s}\cap P_0.$ find conditions for which solution set bounded and stable under $\tilde{E^s_0}$-property. propose an based on interior point method to solve given $A \in \tilde{E^{s}_{0}}.$