Abstract
One of the fundamental questions in nonlinear optimization is how optimization problems behave when the functions defining them change (e.g., by continuous deformation). Recently the study of epi-continuity has somewhat unified the results in this area. Here we show how to localize the concept of epi-continuity, and how to apply these localized ideas to ensure persistence and stability of local optimizing sets. We also show how these conditions follow from known properties of nonlinear programming problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Attouch,Variational Convergence for Functions and Operators (Pitman, Boston, 1984).
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. TammerNon-linear Parametric Optimization (Akademie-Verlag, Berlin-DDR, 1982).
C. Berge,Topological Spaces (Macmillan, New York, 1963).
G. Buttazzo and G. Dal Maso, “Γ-convergence and optimal control problems”,Journal of Optimization Theory and Applications 38 (1982) 385–407.
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.
S. Dolecki, “Semicontinuity in constrained optimization, Part 1: Metric spaces”,Control and Cybernetics 7 (1978) 2, 5–16.
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).
G.H. Greco,Ann. Univ. Ferrara 29 (1983) 153–164.
W.W. Hogan, “Point-to-set maps in mathematical programming”,SIAM Review 15 (1973) 591–603.
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).
S.M. Robinson, “Stability theory for systems of inequalities. Part II: Differentiable nonlinear systems”,SIAM Journal on Numerical Analysis 13 (1976) 497–503.
S.M. Robinson, “Generalized equations and their solutions, Part II: Applications to mathematical programming”,Mathematical Programming Study 19 (1982) 200–221.
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.
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).
T. Zolezzi, “On stability analysis in mathematical programming”,Mathematical Programming Study 21 (1984) 227–242.
Author information
Authors and Affiliations
Additional information
Sponsored by the National Science Foundation under Grant Nos. MCS-8200632, Mod. 2, and DCR-8502202, and by the United States Army under Contract No. DAAG29-80-C-0041 at the University of Wisconsin-Madison. The first version of this paper was written at the International Institute for Applied Systems Analysis, Laxenburg, Austria, during the author's visit there in May and June of 1983. It appeared as IIASA Collaborative Paper CP-84-5.
Rights and permissions
About this article
Cite this article
Robinson, S.M. Local epi-continuity and local optimization. Mathematical Programming 37, 208–222 (1987). https://doi.org/10.1007/BF02591695
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591695