Complementarity problem

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Let LaTeX: F: \mathbb{R}^n \rightarrow \mathbb{R}^n. The complementarity problem (CP) is to find LaTeX: z \ge 0 such that LaTeX: F(z) \ge 0 and LaTeX: F(z)z' = 0. It is complementary because every solution has the property that either LaTeX: z_j=0 or LaTeX: F_j(z)=0 (or both) for each j. The linear complementarity problem (LCP) is when LaTeX: F(z)=Mz+q.

The problem generalizes to allow bounds so that LaTeX: L \le z \le U. Then, LaTeX: F(z) is required to satisfy:

LaTeX: F_j(z) \ge 0 if LaTeX: z_j = L_j

LaTeX: F_j(z) \le 0 if LaTeX: z_j = U_j

LaTeX: F_j(z) = 0 if LaTeX: L_j < z_j < U_j.


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