Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T16:39:45.276Z Has data issue: false hasContentIssue false
  • English
  • Français

The Proximal Subgradient Formula in Banach Space

Published online by Cambridge University Press:  20 November 2018

Philip D. Loewen
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The proximal subgradient formula is a refinement due to Rockafellar of Clarke's fundamental proximal normal formula. It expresses Clarke's generalized gradient of a lower semicontinuous function in terms of analytically simpler proximal subgradients. We use the infinite-dimensional proximal normal formula recently given by Borwein and Strojwas to derive a new version of the proximal subgradient formula in a reflexive Banach space X with Frechet differentiable and locally uniformly convex norm. Our result improves on the one given by Borwein and Strojwas by referring only to the given norm on X.

Keywords

proximal normalsproximal subgradientsgeneralized gradientsClarke's normalcone26B0546B20

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 3 , 01 September 1988, pp. 353 - 361
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Borwein, J. M. and Giles, J. R., The proximal normal formula in Banach space, Trans. Amer. Math. Soc. 302 (1987), pp. 371–381.Google Scholar
2. Borwein, J. M. and Strojwas, H. M., Proximal analysis and boundaries of closed sets in Banach space - Part I: Theory, Can. J. Math. 38 (1986), pp. 431–452.Google Scholar
3. Borwein, J. M. and Strojwas, H. M., Proximal analysis and boundaries of closed sets in Banach space — Part IT. Applications, Can. J. Math., to appear.Google Scholar
4. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), pp. 247–262.Google Scholar
5. Clarke, F. H., Optimization and nonsmooth analysis, New York, John Wiley, 1983.Google Scholar
6. Clarke, F. H., Perturbed optimal control problems, IEEE Trans. Auto. Control AC-31 (1986), pp. 535–542.Google Scholar
7. Clarke, F. H. and Loewen, P. D., The value function in optimal control: sensitivity, controllability, and time-optimality, SIAM J. Control Optim. 24 (1986), pp. 243–263.Google Scholar
8. Clarke, F. H. and Loewen, P. D., State constraints in optimal control: a case study in proximal normal analysis, SIAM J. Control Optim. 25 (1987), pp. 1440–1456.Google Scholar
9. Day, M. M., Normed linear spaces, 3d ed. Berlin: Springer-Verlag, 1973.Google Scholar
10. Diestel, J., Geometry of Banach spaces — Selected topics, Berlin: Springer-Verlag, 1975. Lecture notes in mathematics, p. 485.Google Scholar
11. Gauvin, J., The generalized gradient of a marginal function in mathematical programming, Math, of Oper. Res. 4 (1979), pp. 458–463.Google Scholar
12. Lau, K. S., Almost Chebyshev subsets in reflexive Banach spaces, Indiana Univ. Math. J. 27 (1978), pp. 791–795.Google Scholar
13. Loewen, P. D., The proximal normal formula in Hilbert space, Nonlinear Analysis 11 (1987), pp. 979–995.Google Scholar
14. Rockafellar, R. T., Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math, of Oper. Res. 6 (1981), pp. 424–436.Google Scholar
15. Rockafellar, R. T., Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Prog. Study 17 (1982), pp. 28–66.Google Scholar
16. Rockafellar, R. T., Extensions of subgradient calculus with applications to optimization, Nonlinear Analysis 9 (1985), pp. 665–698.Google Scholar
17. Treiman, J. S., Characterization of Clarke's tangent and normal cones in finite and infinite dimensions, Nonlinear Analysis 7 (1983), pp. 771–783.Google Scholar
18. Treiman, J. S., Clarke's gradients and epsilon-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294(1986), pp. 65–78.Google Scholar