Implicit Complementarity Problem

Firstly, implicit complementarity problems (Abbr. ICP) is introduced by some optimization problems such as linear programming, quadratic programming. Linear complementarity problems and nonlinear complementarity problems are used in those examples and ICP is thus abstracted.ICP is solved by several means in this paper such as unconstrained optimization, constrained optimization and projection operator. The condition that ensures the local optimal points or KKT points to be the solutions of ICP is discussed in unconstrained optimization and constrained optimization respectively. Classic projection operator and a new tengent projection are also used to solve ICP.Under some assumptions, ICP is equivalent to generalized variational inequality (Abbr. GVI). The famous KKM theory is used to solve the existence of the solution of ICP. Later, new results on nonlinear complementarity problem are deduced.