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.)


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