On a regularized variant of the two-step gradient projection method (Q1281208)

The problem of minimization \[ J(u) \to \inf,\quad u\in\mathbf U \tag{1} \] \[ \mathbf U = (u \in\mathbf{U_{0} };\;g_{i}(u) \leq 0, \;i = \overline{1, m};\;g_{i}(u)=0 , i = \overline{m+1 , s}) \tag{2} \] where \(\mathbf U\) is a given convex closed set of a Hilbert space \(H\), functions \(J (u), g_{1}(u), \ldots, g_{s}(u)\) are definite and differentiable in a sense of Fréchet on \(H\), is considered. It is known that the problem (1), (2) is unstable in relation to the perturbations of the initial values of the functions \( J (u)\), \(g_i(u)\) and therefore it is necessary to apply regularization methods. A method of regularization based on the two steps projection-gradient method is proposed in the article. Sufficient conditions for the convergence of the method proposed are investigated.