A three-step regularized method of linearization for solving minimization problems (Q1902856)

Consider a minimization problem \[ J(u)\to \inf,\quad u\in U= \{u\in U_0: g_i(u)\leq 0,\;i= 1,\dots, m\},\tag{1} \] where \(U_0\) is a given convex set in a Hilbert space \(H\). \(J(u)\) and \(g_i(u)\) re Fréchet differentiable functions defined on \(U_0\). \(\langle .,.\rangle\) is the scalar product in \(H\), \(|u|= \langle u, u\rangle^{1/2}\). To solve equation (1) which is unstable against input data perturbations, a method of regularization is proposed. Consider the iteration process as follows: \[ u_{k+ 1}= P_{U_k}[u_k- \beta_k(u_{k- 1}- u_k)- \sigma_k(u_{k- 2}- u_{k- 1})- \gamma_k t'(u_k)],\;k\geq 2,\tag{2} \] where \(u_0, u_1, u_2\in U_0\) are given points. \(t_k'(u)= J_k'(u)+ \alpha_k u\) is an approximate value of the gradient of the Tikhonov function \(T_k(u)= J(u)+ (\alpha_k/2)|u|^2\), \(u\in U_0\), \(P_{U_k}(z)\) is the projection of \(z\) on the set \[ U_k= \{z\in U_0: g_{ik}(u_k)+ \langle g_{ik}'(u_i), z- u_k\rangle\leq \theta(1+ |u_k|^2),\;i= 1,\dots, m\},\tag{3} \] \(\alpha_k\), \(\beta_k\), \(\sigma_k\), \(\gamma_k\) and \(\theta_k\) are parameters. \(g_{ik}(u)\) are known approximations of unknown exact values \(g_i(u)\). \(u_{k+ 1}\) in equations (2-3) is a solution of the equivalent problem \[ 0.5|z- u_k+ \beta_k(u_{k-1}- u_k)+ \sigma_k(u_{k- 2}- u_{k- 1})+ \gamma_k t'(u_k)|^2\to \inf,\quad z\in U_k. \] Sufficient conditions ensuring that a sequence \(\{u_k\}\) generated by equations (2-3) converges in \(H\) to a normal solution of equation (1) are proved.