Robinson, Stephen M. Local epi-continuity and local optimization. (English) Zbl 0623.90078 Math. Program. 37, 208-222 (1987). The theory of epi-convergence has unified many of the approaches and results in stability analysis. This paper shows how to localize certain results found in this area, and how to use these localized results to develop useful criteria for persistence and stability of local minimizers. These criteria are implied by assumptions commonly used in optimization, such as constraint qualifications and second-order sufficient conditions. Reviewer: M.Ying Cited in 28 Documents MSC: 90C31 Sensitivity, stability, parametric optimization 90C30 Nonlinear programming 49M37 Numerical methods based on nonlinear programming Keywords:epi-convergence; stability analysis; stability of local minimizers; constraint qualifications PDF BibTeX XML Cite \textit{S. M. Robinson}, Math. Program. 37, 208--222 (1987; Zbl 0623.90078) Full Text: DOI References: [1] H. Attouch,Variational Convergence for Functions and Operators (Pitman, Boston, 1984). · Zbl 0561.49012 [2] B. Bank, J. Guddat, D. Klatte, B. Kummer and K. TammerNon-linear Parametric Optimization (Akademie-Verlag, Berlin-DDR, 1982). [3] C. Berge,Topological Spaces (Macmillan, New York, 1963). · Zbl 0114.38602 [4] G. Buttazzo and G. Dal Maso, ”\(\Gamma\)-convergence and optimal control problems”,Journal of Optimization Theory and Applications 38 (1982) 385–407. · Zbl 0471.49012 · doi:10.1007/BF00935345 [5] E. De Giorgi and T. Franzoni, ”Su un tipo de convergenza variationale”,Accademia Nazionale dei Lincei, Classe di scienze fisiche, matematiche e naturali, Rendiconti 58 (1975) 842–850. · Zbl 0339.49005 [6] S. Dolecki, ”Semicontinuity in constrained optimization, Part 1: Metric spaces”,Control and Cybernetics 7 (1978) 2, 5–16. · Zbl 0401.90085 [7] S. Dolecki, in: G. Hammer and D. Pallaschke, eds.,Selected Topics in Operations Research and Mathematical Economics, Proceedings, 1983 (Lecture Notes in Economics and Mathematical Systems No. 226, Springer-Verlag, Berlin, 1984). [8] G.H. Greco,Ann. Univ. Ferrara 29 (1983) 153–164. [9] W.W. Hogan, ”Point-to-set maps in mathematical programming”,SIAM Review 15 (1973) 591–603. · Zbl 0256.90042 · doi:10.1137/1015073 [10] J.-P. Penot, ”Continuity properties of performance functions”, in: J.-B. Hiriart-Urruty, W. Oettli and J. Stoer, eds.,Optimization: Theory and Algorithms (Lecture Notes in Pure and Applied Mathematics No. 86, Marcel Dekker, New York, 1983). [11] S.M. Robinson, ”Stability theory for systems of inequalities. Part II: Differentiable nonlinear systems”,SIAM Journal on Numerical Analysis 13 (1976) 497–503. · Zbl 0347.90050 · doi:10.1137/0713043 [12] S.M. Robinson, ”Generalized equations and their solutions, Part II: Applications to mathematical programming”,Mathematical Programming Study 19 (1982) 200–221. · Zbl 0495.90077 [13] S.M. Robinson, ”Local structure of feasible sets in nonlinear programming, Part 1: Regularity”, in: V. Pereyra and A. Reinoza, eds..,Numerical Methods (Lecture Notes in Mathematics No. 1005, Springer-Verlag, Berlin, 1983) pp. 240–251. [14] R.T. Rockafellar and R.J.-B. Wets, ”Variational systems, an introduction”, preprint, Centre de Recherche de Mathématiques de la Décision, Université Paris-IX Dauphine (Paris, 1984). [15] T. Zolezzi, ”On stability analysis in mathematical programming”,Mathematical 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.