A three-step method of linearization for minimization problems (Q1902852)

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 closed set in the Euclidean space \(E^n\). \(J(u)\) and \(g_i(u)\) are convex differentiable functions defined on \(U_0\). \(\langle.,.\rangle\) is the scalar product in \(E^n\), \(|u|= \langle u, u\rangle^{1/2}\). Suppose \(J_*= \inf_{u\in U} J(u)>- \infty\), \(U_*= \{u\in U: J(u)= J_*\}\neq \emptyset\). To solve (1), a method of linearization is proposed as follows: \[ \begin{cases} u_{k+ 1}= P_{U_k} [u_k- \beta_k(u_{k- 1}- u_k)- \sigma_k(u_{k- 2}- u_{k- 1})- \alpha_k J'(u_k)],\;k\geq 2,\\ U_k= \{z\in U_0: g_i(u_k)+ \langle g_i'(u_i), z- u_k\rangle\leq 0,\;i= 1,\dots, m\},\end{cases}\tag{2} \] where \(u_0, u_1, u_2\in U_0\) are given points, \(P_{U_k}(z)\) is the projection of \(z\) onto \(U_k\). \(\alpha_k\), \(\beta_k\) and \(\sigma_k\) are parameters. \(u_{k+ 1}\) in (2) is a solution of the equivalent problem \[ 0.5|z- u_k+ \beta k(u_{k- 1}- u_k)+ \sigma_k(u_{k- 2}- u_{k- 1})+ \alpha_k J'(u_k)|^2\to \inf,\quad z\in U_k.\tag{3} \] Sufficient conditions ensuring that a sequence \(\{u_k\}\) generated by (2) converges to some solution \(u_\infty\in U_*\) of (1) are proved.