Complementarity theory

A complementarity problem is a type of mathematical optimization problem. 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.^[1] In particular for finite-dimensional real vector spaces this means that, if one has vectors X and Y with all nonnegative components (x_i ≥ 0 and y_i ≥ 0 for all ${\displaystyle i}$: in the first quadrant if 2-dimensional, in the first octant if 3-dimensional), then for each pair of components x_i and y_i one of the pair must be zero, hence the name complementarity. e.g. X = (1, 0) and Y = (0, 2) are complementary, but X = (1, 1) and Y = (2, 0) are not. A complementarity problem is a special case of a variational inequality.