# Gradient projection method

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A feasible direction method by projecting the gradient into the working surface, $LaTeX: \{x: Ax=b\}$. Suppose $LaTeX: A$ has full row rank. Then, $LaTeX: P = I - A^T(AA^T)^{-1}A$ projects any vector into the null space of $LaTeX: A$: $LaTeX: APy = (A-A)y = 0$ for all $LaTeX: y \in \mathbb{R}^n$. The form of an iteration is $LaTeX: x' = x + sd$, where $LaTeX: d$ is the projected gradient, $LaTeX: P \nabla f(x)$, and $LaTeX: s$ is determined by line search. Since $LaTeX: Ad=0$, $LaTeX: Ax'=Ax$, thus staying in the working surface. (This extends to nonlinear constraints by using the same correction procedure as the generalized reduced gradient method.)