Convergence results for a class of abstract continuous descent methods

Electronic Journal of Differential Equations

Volumn 2004, Issue , 2004, Pages 1-13

  Aizicovici, Sergiu ^a   Reich, Simeon ^b   Zaslavski, Alexander J ^b  

^a OHIO UNIVERSITY   (United States)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Complete metric space; Descent method; Lipschitzian function; Porous set; Regular vector field

Indexed keywords

EID: 3042694088     PISSN: 10726691     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (18)

1. S. Aizicovici, S. Reich and A. J. Zaslavski, Convergence theorems for continuous descent methods, J. Evol. Equ., to appear.
2. Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2000.
3. H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North Holland, Amsterdam, 1973.
4. H. B. Curry, The method of steepest descent for nonlinear minimization problems, Quart. Appl. Math., Vol. 2 (1944), 258-261.
5. F. S. De Blasi and J. Myjak, Generic flows generated by continuous vector fields in Banach spaces, Adv. Math., Vol. 50 (1983), 266-280.
6. F. S. De Blasi and J. Myjak, Sur la porosité des contractions sans point fixe, C. R. Acad. Sci. Paris, Vol. 308 (1989), 51-54.
7. F. S. De Blasi, J. Myjak and P. L. Papini, Porous sets in best approximation theory, J. London Math. Soc., Vol. 44(1991), 135-142.
8. I. Ekeland, On the variational principle, J. Math. Anal. Appl., Vol. 47 (1974), 324-353.
9. J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993.
10. A. D. Ioffe and A. J. Zaslavski, Variational principles and well-posedness in optimization and calculus of variations, SIAM J. Control Optim., Vol. 38 (2000), 566-581.
11. J.W. Neuberger, Sobolev Gradients and Differential Equations, Lecture Notes in Math. No. 1670, Springer, Berlin, 1997.
12. S. Reich and A. J. Zaslavski, Generic convergence of descent methods in Banach spaces, Math. Oper. Research, Vol. 25 (2000), 231-242.
13. S. Reich and A. J. Zaslavski, The set of divergent descent methods in a Banach space is σ-porous, SIAM J. Optim. Vol. 11 (2001), 1003-1018.
14. S. Reich and A. J. Zaslavski, Porosity of the set of divergent descent methods, Nonlinear Anal., Vol. 47 (2001), 3247-3258.
15. S. Reich and A. J. Zaslavski, Two convergence results for continuous descent methods, Electron. J. Differential Equations, Vol. 2003 (2003), No. 24, 1-11.
16. L. Zajíĉek Porosity and σ-porosity, Real Anal. Exchange, Vol. 13 (1987), 314-350.
17. L. Zajíĉek Small non-σ-porous sets in topologically complete metric spaces, Colloq. Math., Vol. 77 (1998), 293-304.
18. A. J. Zaslavski, Well-posedness and porosity in optimal control without convexity assumptions, Calc. Var., Vol. 13 (2001), 265-293.