What is Complementarity theory?

A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing a function of two vector variables subject to certain requirements which include: that the inner product of the two vectors must equal zero, i.e. they are orthogonal. In particular for finite-dimensional real vector spaces this means that, if one has vectors X and Y with all nonnegative components , then for each pair of components xi and yi one of the pair must be zero, hence the name complementarity. e.g. X =  and Y =  are complementary, but X =  and Y =  are not. A complementarity problem is a special case of a variational inequality.