# Complementarity problem

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 $LaTeX: L_j < z_j < U_j$.