Regularized continuous second-order linearization method for minimization problems with inaccurate source data (Q1281224)

A problem of minimization \[ J(u) \rightarrow \inf, \quad u\in U= \{u\in U_0: g_{i} (u)< 0,\;i= 1,\ldots,m\}, \tag{1} \] where \(U_{0}\) is a given convex bounded set of a Hilbert space \(H\), the functions \(J (u)\), \(g_{1} (u),\ldots, g_{m}(u)\) are defined and differentiable in Fréchet's sense is considered. It is known that the problem (1) is not stable in relation to initial values of the functions \(J (u)\), \(g_{i} (u)\) and to solve the problem methods of regularization should be used. A method of regularization based on a continuous variant of the second-order linearization method is proposed. A regulating operator is constructed and sufficient conditions for convergence of the method are obtained.