Regularized gradient projection method based on set extension (Q1288042)

The problem of minimization \[ J(u)\to \inf,\tag{1} \] \[ u\in U=\{u=(u^1,\dots,u^n)\in U_0, \;g_i(u)\leq 0, \;i=\overline{1,m};\;g_i(u)=0,\;i=\overline {m+1,s}\}, \] is considered, where \(U_0\) is a given set of the Euclidean space \(E^n\), the functions \(J(u)\), \(g_1(u),\dots\), \(g_s(u)\) are defined in \(U_0\). 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 \(J(u), g_i(u)\) and therefore it is necessary to apply some regularization methods. The regularizing method of gradient projection is one of the methods usually used. A variant of the gradient projection method, based on an idea of the extension set is proposed in the article. A regularizing operator is constructed and the convergence of the method proposed is investigated.