Alternating Minimization as Sequential Unconstrained Minimization: A Survey

Journal of Optimization Theory and Applications

Volumn 156, Issue 3, 2013, Pages 554-566

  Byrne, Charles L ^a  

^a University of Massachusetts Lowell   (United States)

Author keywords

Alternating minimization; Optimization; Sequential unconstrained optimization

Indexed keywords

ALGEBRA; MAXIMUM LIKELIHOOD; MAXIMUM PRINCIPLE; OPTIMIZATION;

ALTERNATING MINIMIZATION; AUXILIARY FUNCTIONS; BARRIERS FUNCTIONS; FUNCTION METHODS; FUNCTION VALUES; OBJECTIVE FUNCTIONS; OPTIMISATIONS; SEQUENTIAL UNCONSTRAINED OPTIMIZATION; UNCONSTRAINED MINIMIZATION; UNCONSTRAINED OPTIMIZATION;

ITERATIVE METHODS;

EID: 84874249177     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-012-0134-2     Document Type: Article

References (30)

1. Csiszár, I., Tusnády, G.: Information geometry and alternating minimization procedures. Stat. Decis. Suppl. 1, 205-237 (1984).
2. Rockmore, A., Macovski, A.: A maximum likelihood approach to emission image reconstruction from projections. IEEE Trans. Nucl. Sci. NS-23, 1428-1432 (1976).
3. Shepp, L., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
4. Lange, K., Carson, R.: EM reconstruction algorithms for emission and transmission tomography. J. Comput. Assist. Tomogr. 8, 306-316 (1984).
5. Vardi, Y., Shepp, L. A., Kaufman, L.: A statistical model for positron emission tomography. J. Am. Stat. Assoc. 80, 8-20 (1985).
6. Lange, K., Bahn, M., Little, R.: A theoretical study of some maximum likelihood algorithms for emission and transmission tomography. IEEE Trans. Med. Imaging MI-6(2), 106-114 (1987).
7. Byrne, C.: Iterative image reconstruction algorithms based on cross-entropy minimization. IEEE Trans. Image Process. 2, 96-103 (1993).
8. Byrne, C.: Erratum and addendum to 'Iterative image reconstruction algorithms based on cross-entropy minimization'. IEEE Trans. Image Process. 4, 225-226 (1995).
9. Darroch, J., Ratcliff, D.: Generalized iterative scaling for log-linear models. Ann. Math. Stat. 43, 1470-1480 (1972).
10. Schmidlin, P.: Iterative separation of sections in tomographic scintigrams. Nucl. Med. 9(1), 1-16 (1972).
11. Byrne, C.: Iterative reconstruction algorithms based on cross-entropy minimization. In: Levinson, S. E., Shepp, L. (eds.) Image Models (and their Speech Model Cousins). IMA Volumes in Mathematics and Its Applications, vol. 80, pp. 1-11. Springer, New York (1996).
12. Fiacco, A., McCormick, G.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM Classics in Mathematics. SIAM, Philadelphia (1990).
13. Byrne, C.: Sequential unconstrained minimization algorithms for constrained optimization. Inverse Probl. 24(1), 015013 (2008).
14. Teboulle, M.: Entropic proximal mappings with applications to nonlinear programming. Math. Oper. Res. 17(3), 670-690 (1992).
15. Censor, Y., Zenios, S. A.: Proximal minimization algorithm with D-functions. J. Optim. Theory Appl. 73(3), 451-464 (1992).
16. Censor, Y., Zenios, S. A.: Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, New York (1997).
17. Bauschke, H., Combettes, P., Noll, D.: Joint minimization with alternating Bregman proximity operators. Pac. J. Optim. 2, 401-424 (2006).
18. Kullback, S., Leibler, R.: On information and sufficiency. Ann. Math. Stat. 22, 79-86 (1951).
19. Dempster, A. P., Laird, N. M., Rubin, D. B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 37, 1-38 (1977).
20. McLachlan, G. J., Krishnan, T.: The EM Algorithm and Extensions. Wiley, New York (1997).
21. 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. USSR Comput. Math. Math. Phys. 7, 200-217 (1967).
22. Butnariu, D., Byrne, C., Censor, Y.: Redundant axioms in the definition of Bregman functions. J. Convex Anal. 10, 245-254 (2003).
23. Chen, G., Teboulle, M.: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim. 3, 538-543 (1993).
24. Bauschke, H., Borwein, J.: Joint and separate convexity of the Bregman distance. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Studies in Computational Mathematics, vol. 8, pp. 23-36. Elsevier, Amsterdam (2001).
25. Eggermont, P., LaRiccia, V.: Maximum Penalized Likelihood Estimation. Density Estimation, vol. I. Springer, New York (2001).
26. Byrne, C., Censor, Y.: Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization. Ann. Oper. Res. 105, 77-98 (2001).
27. Censor, Y., Elfving, T.: A multi-projection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221-239 (1994).
28. Bauschke, H., Combettes, P.: Iterating Bregman retractions. SIAM J. Optim. 13, 1159-1173 (2003).
29. Byrne, C., Eggermont, P.: EM Algorithms (2012) in preparation.
30. Byrne, C.: Bregman-Legendre multi-distance projection algorithms for convex feasibility and optimization. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Studies in Computational Mathematics, vol. 8, pp. 87-100. Elsevier, Amsterdam (2001).