Metric subregularity and the proximal point method

Journal of Mathematical Analysis and Applications

Volumn 360, Issue 2, 2009, Pages 681-688

  Leventhal, D ^a  

^a School of Chemical and Molecular Engineering   (United States)

Author keywords

Firmly non expansive mapping; Metric regularity; Metric subregularity; Monotone operator; Proximal point; Randomization; Resolvent

Indexed keywords

EID: 70349376403     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2009.07.012     Document Type: Article

References (34)

1. Aragón Artacho F.J., Dontchev A.L., and Geoffroy M.H. Convergence of the proximal point method for metrically regular mappings. CSVAA 2004-Control Set-Valued Analysis and Applications. ESAIM Proc. vol. 17 (2007), EDP Sci., Les Ulis 1-8
2. Artacho F.J.A., and Geoffroy M.H. Characterization of metric regularity of subdifferentials. J. Convex Anal. 15 2 (2008) 365
3. Bauschke H.H., and Borwein J.M. On the convergence of von Neumann's alternating projection algorithm for two sets. Set-Valued Anal. 1 (1993) 185-212
4. Bauschke H.H., and Borwein J.M. On projection algorithms for solving convex feasibility problems. SIAM Rev. 38 3 (1996) 367-426
5. Bauschke H.H., Borwein J.M., and Lewis A.S. The method of cyclic projections for closed convex sets in Hilbert space. Recent Developments in Optimization Theory and Nonlinear Analysis: Ams/Imu Special Session on Optimization and Nonlinear Analysis, vol. 2. May 24-26, 1995, Jerusalem, Israel (1997), American Mathematical Society
6. Bauschke H.H., Borwein J.M., and Tseng P. Bounded linear regularity strong CHIP and CHIP are distinct properties. J. Convex Anal. 7 2 (2000) 395-412
7. Beck A., and Teboulle M. Convergence rate analysis error bounds for projection algorithms in convex feasibility problems. Optim. Methods Softw. 18 4 (2003) 377-394
8. Combettes P.L. Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53 5 (2004) 475-504
9. Deutsch F. Best Approximation in Inner Product Spaces (2001), Springer-Verlag, New York
10. Dmitruk A.V., Milyutin A.A., and Osmolovskii N.P. Lyusternik's theorem the theory of extrema. Russian Math. Surveys 35 6 (1980) 11-51
11. Dontchev A.L., Lewis A.S., and Rockafellar R.T. The radius of metric regularity. Trans. Amer. Math. Soc. 355 2 (2003) 493-517
12. Dontchev A.L., and Rockafellar R.T. Regularity conditioning of solution mappings in variational analysis. Set-Valued Anal. 12 (2004) 79-109
13. J. Eckstein, Splitting methods for monotone operators with applications to parallel optimization, PhD thesis, Massachusetts Institute of Technology, 1989
14. Graves L.M. Some mapping theorems. Duke Math. J. 17 1 (1950) 111-114
15. Hirstoaga S.A. Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 324 2 (2006) 1020-1035
16. Ioffe A.D. Metric regularity subdifferential calculus. Russian Math. Surveys 55 3 (2000) 501-558
17. Ioffe A.D., and Outrata J.V. On metric calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16 2 (2008) 199-227
18. Kiwiel K.C., and Lopuch B. Surrogate projection methods for finding fixed points of firmly nonexpansive mappings. SIAM J. Optim. 7 4 (1997) 1084-1102
19. Kruger A. Stationarity regularity of set systems. Pac. J. Optim. 1 (2005) 101-126
20. Kruger A.Y. About regularity of collections of sets. Set-Valued Anal. 14 2 (2006) 187-206
21. Lehdili N., and Lemaire B. The barycentric proximal method. Comm. Appl. Nonlinear Anal. 6 (1999) 29-47
22. D. Leventhal, A.S. Lewis, Randomized methods for linear constraints: Convergence rates and conditioning, submitted for publication; available online at: http://arxiv.org/pdf/0806.3015
23. Lewis A.S., Luke D.R., and Malick J. Local convergence for alternating averaged nonconvex projections. Found. Comput. Math. 9 4 (August 2009)
24. Luque F.J. Asymptotic convergence analysis of the proximal point algorithm. SIAM J. Control Optim. 22 (1984) 277
25. Lyusternik L.A. Conditional extrema of functionals. Mat. Sb. 41 3 (1934) 390-401
26. Martinet B. Regularisation d'inequations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4 (1970) 154-159
27. Minty G.J. Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29 3 (1962) 341-346
28. Ngai H.V., and Théra M. Metric inequality subdifferential calculus and applications. Set-Valued Anal. 9 1 (2001) 187-216
29. Pennanen T. Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27 1 (2002) 170
30. Pierra G. Decomposition through formalization in a product space. Math. Program. 28 (1984) 96-115
31. Reich S. A limit theorem for projections. Linear Multilinear Algebra 13 3 (1983) 281-290
32. Rockafellar R.T. Monotone operators the proximal point algorithm. SIAM J. Control Optim. 14 5 (1976) 877-898
33. Rockafellar R.T., and Wets R.J.-B. Variational Analysis (1998), Springer-Verlag, Berlin
34. Solodov M.V., and Svaiter B.F. A hybrid projection-proximal point algorithm. J. Convex Anal. 6 (1999) 59-70