Continuous second-order variable-metric linearization method (Q1290358)

The problem of minimization \[ J(u)\to \inf, \quad u\in U=\{u\in U_0\mid g_i(u)\leq 0, i=1,\dots,l\}, \tag{1} \] is considered, where \(U_0\) is a given convex closed set of a Hilbert space \(H\), the functions \(J(u),g_1(u),\dots,g_l(u)\) are defined and differentiable on \(H\). It is supposed that \[ J_*=\inf_{u\in U}J(u)>-\infty,\quad U_*=\{u\in U:J(u)=J_*\}\neq \emptyset.\tag{2} \] It is known that the problem (1), (2) is unstable in relation to perturbations of the initial data \(J(u), g_i(u),g_i'(u)\) and therefore it is necessary to apply some regularization methods. A continuous linearization method of the second order with a variable metric is proposed in the article. It is noted that the method proposed has some advantage in comparison to known methods.