On regularization of unstable minimization problems
The paper contains some sufficient conditions for the convergence of the Tikhonov regularization method [cf. \textit{F. P. Vasil'ev}, ``Methods for solving extremal problems'' (Russian) (1981; Zbl 0553.49001), \S2.5] for the following optimization problem: (1) \(J(u)\rightarrow\inf (u\in U)\), where \[ U=\{u_ 0\in U_ 0: g_ i(u)\leq 0,\;i=1,...,m;\;g_ i(u)=0,\;i=m+1,...,s\},\leqno (2) \] and \(U_ 0\) is a given subset of some metric space. It is assumed that the problem (1)--(2) satisfies the following condition: there exist constants \(\nu>0\), \(c_ 1\geq0,\ldots,c_ s\geq0\) such that \[ \inf\{J(v): v\in U\}\leq J(u)+\sum^ s_{i=1}c_ i(g^ +_ i(u))^ \nu\hbox{ for all }u\in U_ 0, \] where \(g^ +_ i=\max\{g_ i,0\}\) for \(i=1,\ldots,m\), and \(g^ +_ i=| g_ i|\) for \(i=m+1,\ldots,s\).
scientific article; zbMATH DE number 3556
scientific article