It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements (constraints) which include: that the inner product of the two vectors must equal zero, i.e. they are orthogonal.
A complementarity problem is a special case of a variational inequality.
In 1963 Lemke and Howson showed that, for two person games, computing a Nash equilibrium point is equivalent to an LCP.
In 1968 Cottle and Dantzig unified linear and quadratic programming and bimatrix games.
Since then the study of complementarity problems and variational inequalities has expanded enormously.