Joint and separate convexity of the bregman distance

Studies in Computational Mathematics

Volumn 8, Issue C, 2001, Pages 23-36

  Bauschke, Heinz H ^a   Borwein, Jonathan M ^b  

^a OKANAGAN UNIVERSITY COLLEGE   (Canada)

^b SIMON FRASER UNIVERSITY   (Canada)

Author keywords

Bregman distance; convex function; joint convexity; separate convexity

Indexed keywords

EID: 77956679007     PISSN: 1570579X     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1570-579X(01)80004-5     Document Type: Chapter

References (35)

1. Bauschke H.H., and Borwein J.M. Legendre functions and the method of random Bregman projections. Journal of Convex Analysis 4 (1997) 27-67
2. Bauschke H.H., Noll D., Celler A., and Borwein J.M. An EM-algorithm for dynamic SPECT tomography. IEEE Transactions on Medical Imaging 18 (1999) 252-261
3. Bhatia R. Matrix Analysis (1996), Springer-Verlag
4. Unpublished manuscript. Browein J.M., and Hsu L.C. On the Joint Convexity of the Bregman Disance (1993)
5. Borwein J.M., and Lewis A.S. Convex Analysis and Nonlinear Optimization (2000), Springer-Verlag
6. Bregman L.M. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. U.S.S.R. Computational Mathematics and Mathematical Physics 7 (1967) 200-217
7. Byrne C., and Censor Y. Proximity function minimization and the convex feasibility problem for jointly convex Bregman distances. Technical Report (1998)
8. C. Byrne and Y. Censor. Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback-Leibler Distance Minimization. Technical Report: June 1999, Revised: April 2000, Annals of Operations Research, to appear.
9. Butnariu D., Censor Y., and Reich S. Iterative Averaging of Entropic Projections for Solving Stochastic Convex Feasibility Problems. Computational Optimization and Applications 8 (1997) 21-39
10. Butnariu D., and Iusem A.N. On a proximal point method for convex optimization in Banach speces. Numerical Functional Analysis and Optimization 18 (1997) 723-744
11. Butnariu D., and Iusem A.N. Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization (2000), Kluwer
12. D. Butnariu, A. N. Iusem, and R. S. Burachik. Iterative methods for solving stochasticconvex feasibility problems. To appear in Computational Optimization and Applications.
13. Preprint. Butnariu D., Reich S., and Zaslavski A.J. Asymptotic behaviour of quasinonexpansive mappings (2000)
14. Censor Y., and Zenios S.A. Parallel Optimization (1997), Oxford University Press
15. Centre f.E.a.C.M. Computational Convex Analysis (1997) project at
16. Clarke F.H., Ledyaev Y.S., Stern R.J., and Wolenski P.R. Nonsmooth Analysis and Control Theory (1998), Springer-Verlag
17. Csiszár I., and Tusnády G. Information geometry and alternating minimization procedures. Statistics and Decisions Supplement 1 (1984) 205-237
18. Preprint. Eggermont P.P.B., and LaRiccia V.N. On EM-like algorithms for minimum distance estimation (2000)
19. Horn R.A., and Johnson C.R. Matrix Analysis (1985), Cambridge University Press
20. Iusem A.N. A short convergence proof of the EM algorithm for a specific Poisson model. Revista Brasileira de Probabilidade e Estatistica 6 (1992) 57-67
21. A. N. Iusem. Personal communication.
22. Kiwiel K.C. Free-steering relaxation methods for problems with strictly convex costs and linear constraints. Mathematics of Operations Research 22 (1997) 326-349
23. Kiwiel K.C. Proximal minimization methods with generalized Bregman functions. SIAM Journal on Control and Optimization 35 (1997) 1142-1168
24. Kiwiel K.C. Generalized Bregman projections in convex feasibility problems. Journal of Optimization Theory and Applications 96 (1998) 139-157
25. Lang S. Undergraduate Analysis. Second Edition (1997), Springer-Verlag
26. Lewis A.S. Convex analysis on the Hermitian matrices. SIAM Journal on Optimization 6 (1996) 164-177
27. Preprint. Lewis A.S., and Sendov H.S. Characterization of Twice Differentiable and Twice Continuously Differentiable Spectral Functions (2000)
28. Preprint. Lewis A.S., and Sendov H.S. Quadratic Expansions of Spectral Functions (2000)
29. Lindblad G. Entropy, information and quantum measurements. Communications in Mathematical Physics 33 (1973) 305-322
30. Marshall A.W., and Olkin I. Inequalities: Theory of Majorization and Its Applications (1979), Academic Press
31. Roberts A.W., and Varberg D.E. Convex Functions (1973), Academic Press
32. Rockafellar R.T. Convex Analysis (1970), Princeton University Press
33. Rockafellar R.T., and Wets R.J.-B. Variational Analysis (1998), Springer-Verlag
34. M. V. Solodov and B. F. Svaiter. An inexact hybrid generalized proximal point methodalgorithm and some new results on the theory of Bregman functions. To appear in Mathematics of Operations Research.
35. Stromberg K.R. An Introduction to Classical Real Analysis (1981), Wadsworth