# A multiprojection algorithm using Bregman projections in a product space

- 578 Downloads
- 457 Citations

## Abstract

Generalized distances give rise to generalized projections into convex sets. An important question is whether or not one can use within the same projection algorithm different types of such generalized projections. This question has practical consequences in the area of signal detection and image recovery in situations that can be formulated mathematically as a convex feasibility problem. Using an extension of Pierra's product space formalism, we show here that a multiprojection algorithm converges. Our algorithm is fully simultaneous, i.e., it uses in each iterative step*all* sets of the convex feasibility problem. Different multiprojection algorithms can be derived from our algorithmic scheme by a judicious choice of the Bregman functions which govern the process. As a by-product of our investigation we also obtain blockiterative schemes for certain kinds of linearly constraned optimization problems.

## Keywords

Non-orthogonal projections convex feasibility problem bregman's generalized distance## Subject classification

90 C25 90 C30## Preview

Unable to display preview. Download preview PDF.

## References

- [1]H.H. Bauschke and J.M. Borwein, On projection algorithms for solving convex feasibility problems, Research Report 93-12, Dept. of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada (June 1993).Google Scholar
- [2]A Ben-Israel and T.N.E. Greville,
*Generalized Inverses, Theory and Applications*(Wiley, 1974).Google Scholar - [3]L.M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comp. Math. Math. Phys. 7 (1967) 200–217.Google Scholar
- [4]D. Butnariu and Y. Censor, On the behavior of a block-iterative projection method for solving convex feasibility problems. Int. J. Comp. Math. 34 (1990) 79–94.Google Scholar
- [5]C.L. Byrne, Iterative image reconstruction algorithms based on cross-entropy minimization, IEEE Trans. Image Proc. IP-2 (1993) 96–103.Google Scholar
- [6]Y. Censor, Row-action methods for huge and sparse systems and their applications, SIAM Rev. 23 (1981) 444–464.Google Scholar
- [7]Y. Censor, A.R. De Pierro, T. Elfving, G.T. Herman and A.N. Iusem, On iterative methods for linearily constrained entropy maximization, in:
*Numerical Analysis and Mathematical Modelling*, ed. A. Wakulicz, Banach Center Publications (PWN-Polish Scientific Publishers, Warsaw, Poland, 1990) pp. 145–163.Google Scholar - [8]Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981) 321–353.Google Scholar
- [9]Y. Censor and J. Segman, On block-iterative entropy maximization. J. Inf. Optim. Sci. 8 (1987) 275–291.Google Scholar
- [10]Y. Censor, Parallel application of block-iterative methods in medical imaging and radiation therapy, Math. Progr. 42 (1988) 307–325.Google Scholar
- [11]Y. Censor and S.A. Zenios, Proximal maximization algorithm with D-functions, J. Optim. Theory Appl. 73 (1992) 451–464.Google Scholar
- [12]P.L. Combettes, The foundations of set theoretic estimation, Proc. IEEE 81 (1993) 182–208.Google Scholar
- [13]I. Csiszár, Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Statist. 19 (1991) 2032–2066.Google Scholar
- [14]J.N. Darroch and D. Ratcliff, Generalized iterative scaling for log-linear models, Ann. Math. Statist. 43 (1972) 1470–1480.Google Scholar
- [15]A.R. De Pierro and A.N. Iusem, A relaxed version of Bregman's method for convex programming, J. Optim. Theory Appl. 51 (1986) 421–440.Google Scholar
- [16]J. Eckstein, Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Math. Oper. Res. 18 (1993) 202–226.Google Scholar
- [17]P.P.B. Eggermont, G.T. Herman and A. Lent, Iterative algorithms for large partitioned linear systems, with applications to image reconstruction, Lin. Alg. Appl. 40 (1981) 37–67.Google Scholar
- [18]P.P.B. Eggermont, Multiplicative iterative algorithms for convex programming, Lin. Alg. Appl. 130 (1990) 25–42.Google Scholar
- [19]T. Elfving, Block-iterative methods for consistent and inconsistent linear equations, Numer. Math. 35 (1980) 1–12.Google Scholar
- [20]T. Elfving, On some methods for entropy maximization and matrix scaling, Lin. Alg. Appl. 34 (1980) 321–339.Google Scholar
- [21]T. Elfving, An algorithm for maximum entropy image reconstruction from noisy data, Math. Comp. Mod. 12 (1989) 729–745.Google Scholar
- [22]L.G. Gubin, B.T. Polyak and E.V. Raik, The method of projection for finding the common point of convex sets, USSR Comp. Math. Math. Phys. 7 (1967) 1–24.Google Scholar
- [23]S.P. Han, A successive projection method, Math. Progr. 40 (1988) 1–14.Google Scholar
- [24]N.E. Hurt,
*Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction*(Kluwer Academic, Dordrecht, The Netherlands, 1989).Google Scholar - [25]A.N. Iusem, On dual convergence and the rate of primal convergence of Bregman's convex programming method, SIAM J. Optim. 1 (1991) 401–423.Google Scholar
- [26]A.N. Iusem, and A.R. De Pierro, On the convergence of Han's method for convex programming with quadratic objective, Math. Progr. 52 (1991) 265–284.Google Scholar
- [27]L.K. Jones and C.L. Byrne, General entropy criteria for inverse problems, with application to data compression, pattern classification and cluster analysis, IEEE Trans. Inf. Theory IT-36 (1990) 23–30.Google Scholar
- [28]K.C. Kiwiel, Block-iterative surrogate projection methods for convex feasibility problems, Technical Report, Systems Research Inst., Polish Academy of Sciences, Newelska 6, 01-447, Warsaw, Poland (December 1992).Google Scholar
- [29]T. Kotzer, N. Cohen and J. Shamir, Extended and alternative projections onto convex sets: theory and applications, Technical Report No. EE 900, Dept. of Electrical Engineering, Technion, Haifa, Israel (November 1993).Google Scholar
- [30]G. Pierra, Decomposition through formalization in a product space. Math. Progr. 28 (1984) 96–115.Google Scholar
- [31]R.T. Rockafellar,
*Convex Analysis*(Princeton University Press, Princeton, NJ 1970).Google Scholar - [32]H. Stark (ed.),
*Image Recovery: Theory and Applications*(Academic Press, Orlando, FL, 1987).Google Scholar - [33]A.E. Taylor,
*Introduction to Functional Analysis*(Wiley, New York, 1967).Google Scholar - [34]M. Teboulle, Entropic proximal mappings with applications to nonlinear programming, Math. Oper. Res. 17 (1992) 670–690.Google Scholar