The set of divergent descent methods in a Banach space IS σ-porous

SIAM Journal on Optimization

Volumn 11, Issue 4, 2001, Pages 1003-1018

  Reich, Simeon ^a   Zaslavski, Alexander J ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Banach space; Complete metric space; Convex function; Descent method; Generic property; Iterative process; Porous set

Indexed keywords

EID: 0035603910     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623400370357     Document Type: Article

References (22)

1. Y.I ALBER, A.N. IUSEM, AND M.V. SOLODOV, Minimization of nonsmooth convex functional in Banach spaces, J. Convex Anal., 4 (1997), pp. 235-255.
2. A. BEN-TAL AND M. ZIBULEVSKY, Penalty/barrier multiplier methods for convex programming problems, SIAM J. Optim., 7 (1997), pp. 347-366.
3. Y. BENYAMINI AND J. LINDENSTRAUSS, Geometric Nonlinear Functional Analysis, AMS, Providence, RI, 2000.
4. R. CORREA AND C. LEMARÉCHAL, Convergence of some algorithms for convex minimization, Math. Programming, 62 (1993), pp. 261-275.
5. F.S. DE BLASI AND J. MYJAK, Generic flows generated by continuous vector fields in Banach spaces, Adv. in Math., 50 (1983), pp. 266-280.
6. F.S. DE BLASI AND J. MYJAK, On a generalized best approximation problem, J. Approx. Theory, 94 (1998), pp. 54-72.
7. R. DEVILLE AND J. REVALSKI, Porosity of ill-posed problems, Proc. Amer. Math. Soc., 128 (2000), pp. 1117-1124.
8. A.L. DONTCHEV AND T. ZOLEZZI, Well-Posed Optimization Problems, Lecture Notes in Math. 1543, Springer, Berlin, 1993.
9. M.S. GOWDA AND M. TEBOULLE, A comparison of constraint qualifications in infinite-dimensional convex programming, SIAM J. Control Optim., 28 (1990), pp. 925-935.
10. J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993.
11. A.D. IOFFE AND A.J. ZASLAVSKI, Variational principles and well-posedness in optimization and calculus of variations, SIAM J. Control Optim., 38 (2000), pp. 566-581.
12. D. PREISS AND L. ZAJICEK, Fréchet differentiation of convex functions in a Banach space with a separable dual, Proc. Amer. Math. Soc., 91 (1984), pp. 202-204.
13. S. REICH AND A.J. ZASLAVSKI, Convergence of generic infinite products of nonexpansive and uniformly continuous operators, Nonlinear Anal., 36 (1999), pp. 1049-1065.
14. S. REICH AND A.J. ZASLAVSKI, On the minimization of convex functionals, in Calculus of Variations and Differential Equations, Res. Notes Math. Ser. 410, CRC Press, Boca Raton, FL, 1999, pp. 200-209.
15. S. REICH AND A.J. ZASLAVSKI, Generic convergence of descent methods in Banach spaces, Math. Oper. Res., 25 (2000), pp. 231-242.
16. J. REVALSKI, Generic properties concerning well-posed optimization problems, C. R. Acad. Bulg. Sci., 38 (1985), pp. 1431-1434.
17. L. ZAJICEK, Porosity and σ-porosity, Real Anal. Exchange, 13 (1987), pp. 314-350.
18. L. ZAJICEK, Products of non-σ-porous sets and Foran systems, Atti Sem. Mat. Fis. Univ. Modena, 44 (1996), pp. 497-505.
19. L. ZAJICEK, Small non-σ-porous sets in topologically complete metric spaces, Colloq. Math., 77 (1998), pp. 293-304.
20. A.J. ZASLAVSKI, Optimal programs on infinite horizon 1 and 2, SIAM J. Control Optim., 33 (1995), pp. 1643-1686.
21. A.J. ZASLAVSKI, Dynamic properties of optimal solutions of variational problems, Nonlinear Anal., 27 (1996), pp. 895-932.
22. A.J. ZASLAVSKI, Existence of solutions of optimal control problems for a generic integrand without convexity assumptions, Nonlinear Anal., 43 (2001), pp. 339-361.