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Regularized gradient projection method based on set extension. (English. Russian original) Zbl 0917.49027

Mosc. Univ. Comput. Math. Cybern. 1997, No. 3, 6-10 (1997); translation from Vestn. Mosk. Univ., Ser. XV 1997, No. 3, 6-9 (1997).
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.

MSC:

90C52 Methods of reduced gradient type
49J35 Existence of solutions for minimax problems
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