Metric Projections and the Gradient Projection Method in Banach Spaces
R. R. Phelps · SIAM Journal on Control and Optimization · 1985
Let $P_C $ denote the metric projection onto a closed convex subset C of a smooth, rotund and reflexive Banach space E. With an additional hypothesis on E, it is shown that $P_C $ has directional derivatives in every direction at every point of C (a result which is well known for Hilbert space). This is used to prove that in such a reflexive space any cluster point for the gradient projection method (using Cauchy’s steplength) is a constrained stationary point.