Projected gradient processes of the Goldstein–Levitin–Polyak type are considered for constrained minimization problems, $\min _\Omega F$, with $\Omega $ a convex set in a Hilbert space X and $F:X \to \mathbb{R}^1 $ a differentiable functional. Global and local convergence theorems are established for a large class of these processes, including those generated with implicit step length rules proposed by Bertsekas and Goldstein. In this analysis, traditional uniform strong positivity conditions on the Hessian $\nabla ^2 F$ are replaced by weaker pseudoconvexity conditions and growth conditions on F. When F has a unique minimizes in $\Omega $, convergence rates are shown to depend on how rapidly the function $\gamma (\sigma ) = \inf \{ r = F(x) - F(\xi )\mid x . \in \Omega \| {x - \xi } \| \geqq \sigma \} $ grows with increasing $\sigma > 0$. If $\gamma (\sigma ) \geqq B\sigma ^\nu $ for some $B > 0$, the processes $\{ F_n \} $ in question converge to $\inf F$ like $O(n^{{{ - \nu } / {(\nu - 2)}}} )$, linear...