Subgradient method for minimization of convex functionals and some efficiency bounds (Q1264094)

Consider the problem of minimizing a finite convex functional f on a real Hilbert space E. Let \(\partial f(y)\) be the subdifferential of the functional f at the point \(y\in E\), i.e., \[ \partial f(y)=\{q\in E:\quad f(x)-f(y)\geq <q,x-y>,\quad \forall x\in E\}. \] We propose a solution method for this problem which is conceptually close to the methods of \textit{V. F. Dem'yanov} and \textit{L. V. Vasil'ev} [``Nondifferentiable optimization'' (1985; Zbl 0593.49001) (For a review of the 1981 Russian original see Zbl 0559.49001)] and \textit{C. Lemarechal} [Inform. Processing 74, Proc. IFIP Congr. 74, Stockholm, 552-556 (1974; Zbl 0297.65041)] and estimate its rate of convergence.