Roubíček, Tomáš A generalized solution of a nonconvex minimization problem and its stability. (English) Zbl 0626.49013 Kybernetika 22, 289-298 (1986). Author’s abstract: “It is well known that the set of the solutions of a minimization problem on an infinite-dimensional space X is not stable with respect to a perturbation of the minimized function. Here a generalized solution is defined as an element of a suitable completion of X. A necessary and sufficient condition for the completion of X to guarantee the stability of the set of the generalized solutions is given. It is shown that the generalized solution can be considered as a certain minimizing filter on X, which generalizes the notion of the minimizing sequence.” Reviewer: J.F.Bonnans Cited in 1 ReviewCited in 2 Documents MSC: 49K40 Sensitivity, stability, well-posedness 49J27 Existence theories for problems in abstract spaces 90C48 Programming in abstract spaces Keywords:minimization problem on an infinite-dimensional space; generalized solution; stability; minimizing filter PDF BibTeX XML Cite \textit{T. Roubíček}, Kybernetika 22, 289--298 (1986; Zbl 0626.49013) Full Text: EuDML References: [1] W. Alt: Lipschitzian perturbation of infinite optimization problems. Mathematical Programming with Data Perturbations II (A. V. Fiacco, Marcel Dekker, Inc., New York-Basel 1983, pp. 7-22. [2] N. Bourbaki: General Topology. Hermann, Paris 1966. · Zbl 0301.54002 [3] Á. Császár: General Topology. Akadémiai Kiadó, Budapest 1978. [4] I. Ekeland, R. Teman: Convex Analysis and Variational Problems. North-Holland, Amsterdam 1976. [5] L. Gillman, M. Jerison: Rings of Continuous Functions. Second edition. Springer-Verlag, Berlin-Heidelberg-New York 1976. · Zbl 0327.46040 [6] E. G. Golshtein: Theory of Convex Programming. AMS, Transl. of Math. Monographs, Vol. 36, Providence, R. I. 1972. [7] A. D. Ioffe, V. M. Tihomirov: Extension of variational problems. Trudy Moskov. Mat. Obšč. (Trans. Moscow Math. Soc.) 18 (1968), 207-273. · Zbl 0194.13902 [8] E. Polak, Y. Y. Wardi: A study of minimizing sequences. SIAM J. Control Optim. 22 (1984), 599-609. · Zbl 0553.49017 · doi:10.1137/0322036 [9] T. Zolezzi: On stability analysis in mathematical programming. Math. Programming Study 21 (1984), 227-242. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.