Regularized continuous variable-metric method of minimization with inaccurate source data (Q1281239)

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