This article is about mathematical programming. For other uses, see Complementarity (disambiguation).
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 (xi ≥ 0 and yi ≥ 0 for all : in the first quadrant if 2-dimensional, in the first octant if 3-dimensional), then for each pair of components xi and yi 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.
^Billups, Stephen; Murty, Katta (2000). "Complementarity Problems". Journal of Computational and Applied Mathematics. 124 (1–2): 303–318. Bibcode:2000JCoAM.124..303B. doi:10.1016/S0377-0427(00)00432-5.
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