Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-29T16:01:01.023Z Has data issue: false hasContentIssue false
  • English
  • Français

Smoothing approximations to nonsmooth optimization problems

Published online by Cambridge University Press:  17 February 2009

X.Q. Yang
X.Q. Yang
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, W.A. 6009, Australia.
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.

We study certain types of composite nonsmooth minimization problems by introducing a general smooth approximation method. Under various conditions we derive bounds on error estimates of the functional values of original objective function at an approximate optimal solution and at the optimal solution. Finally, we obtain second-order necessary optimality conditions for the smooth approximation prob lems using a recently introduced generalized second-order directional derivative.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 3 , January 1995 , pp. 274 - 285
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Ben-Tal, A. and Teboulle, M., “A smoothing technique for nondifferentiable optimization problems”, in Optimization - Fifth French-German Conference (ed. Dolecki, S.), Volume 1405 of Lecture Notes in Math., (Springer-Verlag, New York, 1989) 1–13.Google Scholar
[2]Ben-Tal, A., Teboulle, M. and Yang, W. H., “A least-square method for a class of nonsmooth minimization problems with applications in plasticity”, Applied. Math. Optimiz. 24 (1991) 272–288.CrossRefGoogle Scholar
[3]Bertsekas, D. P., “Nondifferentiable optimization via approximation”, Math. Prog. Study 3 (1975) 1–25.CrossRefGoogle Scholar
[4]Bertsekas, D. P., Constrained optimization and lagrange multiplier methods (Academic Press, New York, London, 1982).Google Scholar
[5]Clarke, F. H., Optimization and nonsmooth analysis (John Wiley, New York, 1983).Google Scholar
[6]El-Attar, R. A., Vidyasagar, M. and Dutta, R. K., “An algorithm for l 1-norm minimization with application to nonlinear l 1 approximation”, SIAMJ. Numer. Anal. 16 (1979) 70–86.CrossRefGoogle Scholar
[7]Fletcher, R., Practical methods of optimization (John Wiley, New York, 1987).Google Scholar
[8]Holmes, R. B., A course on optimization and best approximation (Springer-Verlag, New York, 1972).CrossRefGoogle Scholar
[9]Ioffe, A. D., “Necessary and sufficient conditions for a local minimum, 2: conditions of Levitin-Milijutin-Osmoloviskii type”, SIAM J. Control Optimiz. 17 (1979) 245–250.CrossRefGoogle Scholar
[10]Jennings, L. S. and Teo, K. L., “A computational algorithm for functional inequality constrained optimization problems”, Automatica 26 (1990) 371–375.CrossRefGoogle Scholar
[11]Jeyakumar, V., “Composite nonsmooth programming with Gâteaux differentiability”, SIAM J. Optimiz. 1 (1991) 30–41.CrossRefGoogle Scholar
[12]Polyak, R. A., “Smooth optimization methods for minimax problems”, SIAM J. Control Optimiz. 26 (1988) 1274–1286.CrossRefGoogle Scholar
[13]Teo, K. L. and Goh, C.J., “On constrained optimization problems with nonsmooth cost functionals”, Applied Math. Optimiz. 18 (1988) 181–190.CrossRefGoogle Scholar
[14]Teo, K. L., Rehbock, V. and Jennings, L. S., “A new computational algorithm for functional inequality constrained optimization problems”, Automatica 29 (1993) 789–792.CrossRefGoogle Scholar
[15]Yang, X. Q., “Second-order conditions in C 1,1 optimization with applications”, Numer. Funct. Anal. Appl. 14 (5&6) (1993) 621–632.CrossRefGoogle Scholar
[16]Yang, X. Q., “Generalized second-order directional derivatives and optimality conditions”, Non-linear Analysis — TMA (to appear).Google Scholar
[17]Yang, X. Q. and Jeyakumar, V., “Generalized second-order directional derivatives and optimization with C 1,1 functions”, Optimization 26 (1992) 165–185.CrossRefGoogle Scholar