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- W2070815098 abstract "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 Bsigma ^nu $ for some $B > 0$, the processes ${ F_n } $ in question converge to $inf F$ like $O(n^{{{ - nu } / {(nu - 2)}}} )$, linearly, superlinearly, or in finitely many steps according to whether $nu > 2$, $nu = 2$, $2 > nu > 1$, or $nu = 1$. The growth properties of $gamma (sigma )$ are in turn dependent upon the structure of F, $Omega $ and the norm on X. Close connections also exist here with a hierarchy of extremal types constructed in a recent study of conditional gradient algorithms, and with long-standing notions of singularity for constrained optimal control problems and unconstrained minimization problems on $mathbb{R}^n $." @default.
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- W2070815098 date "1981-05-01" @default.
- W2070815098 modified "2023-09-25" @default.
- W2070815098 title "Global and Asymptotic Convergence Rate Estimates for a Class of Projected Gradient Processes" @default.
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- W2070815098 doi "https://doi.org/10.1137/0319022" @default.
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