Hostname: page-component-f554764f5-246sw Total loading time: 0 Render date: 2025-04-08T17:39:19.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate subdifferentials and applications 3: the metric theory

Published online by Cambridge University Press:  26 February 2010

A. D. Ioffe
A. D. Ioffe
Affiliation:
Department of Mathematics, Technion, Haifa, Israel.
Get access

Extract

This is the final paper of the series of three papers under the same title. The finite dimensional theory developed in the first of them 7 contains first of all:

(a) a calculus having among its consequences the calculi of convex subdifferentials and generalized gradients of Clarke (henceforth sometimes abbreviated C.g.g.) in the most general form which is partly due to the fact that in a finite dimensional space

for any convex function f and

for any SX (A means approximate, C means Clarke); and (b) a theorem stating that approximate subdifferentials are minimal (as sets) among all possible subdifferentials satisfying one or another set of conditions (usually very natural).

Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 1 - 38
Copyright
Copyright University College London 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Clarke, F. H.. Generalized gradients and applications. Trans. Amer. Math. Soc., 205 (1975), 247262.CrossRefGoogle Scholar
2.Clarke, F. H.. Optimization and Nonsmooth Analysis (Wiley-Interscience, 1983)Google Scholar
3.Dolecki, S.. Tangency and differentiation: some applications of convergence theory. Ann. Math. Pure Appl., 130 (1982), 235255CrossRefGoogle Scholar
4.Ekeland, I.. Nonconvex minimization problems. Bull. Amer. Math. Soc., 1 (1979), 443474CrossRefGoogle Scholar
5.Frankowska, H.. Inclusions adjointes associes aux trajectoires minimales. CRAS, 297 (1983), 461464.Google Scholar
6.Ioffe, A. D.. Sous-differentielles approachees de fonctions numeriques. CRAS, 292 (1981), 675678.Google Scholar
7.Ioffe, A. D.. Approximate subdifferentials and applications 1. The finite dimensional theory. Trans. Amer. Math. Soc., 281 (1984), 289316Google Scholar
8.Ioffe, A. D.. Approximate subdifferentials and applications 2. Functions on locally convex spaces. Mathematika, 33 (1986), 111128CrossRefGoogle Scholar
9.Ioffe, A. D.. Approximate subdifferentials of nonconvex functions. Cahier 8120, CERE MADE, Univ. Paris IX Dauphine, 1981.Google Scholar
10.Ioffe, A. D.. Nonsmooth analysis: differential calculus of nondifferentiable mappings. Tram. Amer. Math. Soc., 266 (1981), 156.CrossRefGoogle Scholar
11.Kruger, A. Ya.. Properties of generalized differentials. Siberian Math. J., (1983), 822832.CrossRefGoogle Scholar
12.Kruger, A. Ya. and Mordukhovich, B. Sh.. Extremal points and Euler equations in nonsmooth optimization problems. Doklady Akad. Nauk BSSR, 24 (1980), 684687.Google Scholar
13.Lebourg, G.. Valeur moyenne pour gradient generalise. CRAS, 281 (1975), 795798.Google Scholar
14.Mordukhovich, B. Sh.. Maximum principle in the optimal time control problem with nonsmooth constraints. Prikl. Matem. Mekh., 40 (1976), 10141023.Google Scholar
15.Mordukhovich, B. Sh.. Nonsmooth analysis with nonconvex generalized differentials and conjugate mappings. Dokl. Akad. Nauk BSSR, 28 (1984), 976979.Google Scholar
16.Rockafellar, R. T.. Directionally Lipschitzian functions and subdifferential calculus. Proc. London Math. Soc., 39 (1979), 331355.CrossRefGoogle Scholar
17.Rockafellar, R. T.. Generalized directional derivatives and subgradients of nonconvex functions. Canadian J. Math., 32 (1980), 237280.CrossRefGoogle Scholar
18.Rockafellar, R. T.. Extension of subgradient calculus with application to optimization. Nonlinear Anal. Theory, Methods, Appl, 9 (1985), 66569.CrossRefGoogle Scholar
19.Treiman, J. S., Clarke's gradients and epsilon-subgradients in Banach spaces. Trans. Amer. Math. Soc., 294 (1986), 6578.Google Scholar
20.Hagler, J. and Sullivan, F.. Smoothness and weak sequential compactness. Proc Amer. Math. Soc., 78 (1980), 497503.Google Scholar