Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization

Set-Valued Analysis

Volumn 16, Issue 7-8, 2008, Pages 899-912

  Maingé, Paul Emile ^a  

^a UNIVERSITÉ DES ANTILLES ET DE LA GUYANE   (France)

Author keywords

Convex minimization; Nonsmooth optimization; Projected gradient method; Viscosity method

Indexed keywords

EID: 58149142982     PISSN: 09276947     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11228-008-0102-z     Document Type: Article

References (27)

1. Y.I. Albert 1983 Recurrence relations and variational inequalities Soviet Mathematics, Doklady 27 511 517
2. Y.I. Albert A.N. Iusem 2001 Extension of subgradient techniques for nonsmooth optimization in a Banach space Set-Valued Anal. 9 315 335
3. Y.I. Albert A.N. Iusem M.V. Solodov 1998 On the projected subgradient method for nonsmooth convex optimization in a Hilbert space Math. Program. 81 23 35
4. H.H. Bauschke P.L. Combettes 2001 A weak-to-strong convergence principle for Fejer monotone methods in Hilbert space Math. Oper. Res. 26 248 264
5. L. Bello M. Raydan 2005 Preconditioned spectral projected-gradient method on convex sets J. Comput. Math. 23 225 232
6. D.P. Bertsekas E.M. Gafni 1982 Projection methods for variational inequalities with applications to the traffic assignment problem Math. Program. Stud. 17 139 159
7. D.P. Bertsekas 1976 On the Goldstein-Levitin-Polyak gradient projection method IEEE Trans. Automat. Contr. AC-21 2 174 184
8. C.L. Byrne 2004 A unified treatment of some iterative algorithms in signal processing and image reconstruction Inverse Problems 18 441 453
9. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM Publications, Philadelphia (1983)
10. R. Correa C. Lemaréchal 1993 Convergence of some algorithms for convex minimization Math. Program. 62 261 275
11. Ekeland, I., Themam, R.: Convex analysis and variational problems. In: Classic in Applied Mathematics, p. 28. SIAM, Philadelphia (1999)
12. Y.M. Ermoliev 1966 Methods for solving nonlinear extremal problems Cybernet. 2 1 17
13. W.W. Hager S. Park 2004 The gradient projection method with exact line search J. Glob. Optim. 30 103 118
14. B. Halpern 1967 Fixed points of nonexpanding maps Bull. Amer. Math. Soc. 73 957 961
15. A.N. Iusem 2003 On the convergence properties of the projected gradient method for convex optimization Comput. Appl. Math. 22 1 37 52
16. E.N. Khobotov 1987 A modification of the extragradient method for the solution of variational inequalities and some optimization problems Zh. Vychisl. Mat. Mat. Fiz. 27 1462 1473
17. G.M. Korpelevich 1976 The extragradient method for finding saddle points and other problems Matecon 12 747 756
18. P. Marcotte 1991 Applications of Khobotov's algorithm to variational and network equlibrium problems Inform. Syst. Oper. Res. 29 258 270
19. P.E. Maingé A. Moudafi 2007 Strong convergence of an iterative method for hierarchical fixed-point problems Pacific J. Optim. 3 3 529 538
20. A. Moudafi 2000 Viscosity approximations methods for fixed point problems J. Math. Anal. Appl. 241 46 55
21. N. Nadezhkina W. Takahashi 2006 Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings SIAM J. Optim. 16 4 1230 1241
22. M.V. Solodov P. Tseng 1996 Modified projection methods for monotone variational inequalities SIAM J. Control Optim. 34 5 1814 1834
23. M.V. Solodov S.K. Zavriev 1998 Error stability properties of generalized gradient-type algorithms J. Optim. Theory Appl. 98 663 680
24. M.V. Solodov 1999 A new projection method for variational inequality problems SIAM J. Control Optim. 37 3 756 776
25. N. Xiu C. Wang L. Kong 2007 A note on the gradient projection method with exact stepsize rule J. Comput. Math. 25 2 221 230
26. I. Yamada N. Ogura 2004 Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings Numer. Funct. Anal. Optim. 25 7-8 619 655
27. L.C. Zeng J.C. Yao 2006 Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems Taiwan. J. Math. 10 5 1293 1303