The linear complementarity problem provides a collective framework for many optimization issues. The method proposed by Lemke and Howson in order to enumerate an equilibrium point related to strategies of a bi-matrix game, modified by Lemke called Lemke method with a purpose of solving the problem with linear complementarity condition contributed necessarily to an expansion for the linear complementarity theory. Expanding the applicability from Lemke method to larger classes of matrices has been investigated by a large number of scientists. The formulation of linear complementarity for solving structured stochastic games establishes alternative proof of order field property. The solution set connected with the variational inequality problems and the solution set in connection with related complementarity problem with linear structure are identical. Several views to complementarity problem are proposed to accommodate more real world problems and to diversify the field of application. The non-linear complementarity problem, semi-definite complementarity problem and tensor complementarity problem are the areas developed based on the complementarity issue. In this section, the importance of complementarity in optimization theory is discussed, and it is shown that the matrix types establish an important topic in this context.