On weakly abnormal extremal problems with restrictions (Q2719471)

The authors study the problem of minimization with restrictions of the equality type NEWLINE\[NEWLINE \begin{cases} f(x)\to\min\\ F(x) = 0\end{cases}, NEWLINE\]NEWLINE where \(X\) is the Hilbert space, \(Y\) is the Euclidean \(n\)-dimensional space, \(f: X\to R\) and the mapping \(F:X\to Y\) is assumed to be twice continuously Frechét differentiable. The authors establish the conditions of strictly local minimum for the perturbed problem NEWLINE\[NEWLINE \begin{cases} f_\varepsilon(x) = f(x) + \varepsilon\|x-x_0\|^2\to\min\\ \vspace{.03 in} F_\varepsilon(x) = F(x) + \varepsilon\|x-x_0\|^2\bar z = 0 \end{cases} NEWLINE\]NEWLINE at point \(x_0\). The relationship is studied between the necessary conditions from [\textit{A. V. Arutyunov}, Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 1, 37-62 (1996; Zbl 0884.49018); Itogi Nauki Tekh., Ser. Mat. Anal. 27, 147-235 (1989; Zbl 0726.49015)] and the sufficient conditions of local minimum.