Robust stopping criteria for Dykstra's algorithm

SIAM Journal on Scientific Computing

Volumn 26, Issue 4, 2005, Pages 1405-1414

  Birgin, Ernesto G ^a   Raydan, Marcos ^b  

^a UNIVERSITY OF SÃO PAULO   (Brazil)

^b FACULTAD DE CIENCIAS   (Venezuela)

Author keywords

Alternating projection methods; Convex optimization; Dykstra's algorithm; Stopping criteria

Indexed keywords

OPTIMIZATION; PROBLEM SOLVING; SET THEORY;

ALTERNATING PROJECTION METHODS; CONVEX OPTIMIZATION; DYKSTRA'S METHOD; STOPPING METHODS;

ALGORITHMS;

EID: 23944458167     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/03060062X     Document Type: Article

References (30)

1. H. H. BAUSCHKE AND J. M. BORWEIN, Dykstra's alternating projection algorithm for two sets, J. Approx. Theory, 79 (1994), pp. 418-443.
2. J. P. BOYLE AND R. L. DYKSTRA, A method for finding projections onto the intersection of convex sets in Hilbert spaces, Lecture Notes in Statist., 37 (1986), pp. 28-47.
3. R. BRAMLEY AND A. SAMEH, Rout projection methods for large nonsymmetric linear systems, J. Sci. Statist. Comp., 13 (1992), pp. 168-193.
4. L. M. BREGMAN, Y. CENSOR, AND S. REICH, Dykstra's algorithm as the nonlinear extension of Bregman's optimization method, J. Convex Anal., 6 (1999), pp. 319-333.
5. Y. CENSOR AND G. T. HERMAN, On some optimization techniques in image reconstruction from projections, Appl. Numer. Math., 3 (1987), pp. 365-391.
6. P. L. COMBETTES, The convex feasibility problem in image recovery, Advances in Imaging and Electron Physics, 95 (1996), pp. 155-270.
7. F. DEUTSCH, Best Approximation in Inner Product Spaces, Springer-Verlag, New York, 2001.
8. F. DEUTSCH AND H. HUNDAL, The rate of convergence of Dykstra's cyclic projections algorithm: The polyhedral case, Numer. Funct. Anal. Optim., 15 (1994), pp. 537-565.
9. F. DEUTSCH, Dykstra's cyclic projections algorithm: The rate of convergence. Approximation Theory, Wavelets, and Applications, S. P. Singh, A. Carbone, and B. Watson, eds., Kluwer Academic Publishers, Dordrecht, Netherlands, 1995, pp. 87-94.
10. R. L. DYKSTRA, An algorithm for restricted least squares regression, J. Amer. Statist. Assoc., 78 (1983), pp. 837-842.
11. M. G. EBERLE AND M. C. MACIEL, Finding the closest Toeplitz matrix, Comput. Appl. Math., 22 (2003), pp. 1-18.
12. R. ESCALANTE AND M. RAYDAN, Dykstra's algorithm for a constrained least-squares matrix problem, Numer. Linear Algebra Appl., 3 (1996), pp. 459-471.
13. R. ESCALANTE AND M. RAYDAN, Dykstra's algorithm for constrained least-squares rectangular matrix problems, Comput. Math. Appl., 35 (1998), pp. 73-79.
14. N. GAFFKE AND R. MATHAR, A cyclic projection algorithm via duality, Metrika, 36 (1989), pp. 29-54.
15. U. M. GARCIA-PALOMARES, Preconditioning projection methods for solving algebraic linear systems, Numer. Algorithms, 21 (1999), pp. 157-164.
16. W. GLUNT, An alternating projections method for certain linear problems in a Hilbert space, IMA J. Numer. Anal., 15 (1995), pp. 291-305.
17. W. GLUNT, T. L. HAYDEN, S. HONG, AND J. WELLS, An alternating projection algorithm for computing the nearest Euclidean distance matrix, SIAM J. Matrix Anal. Appl., 11, (1990), pp. 589-600.
18. W. GLUNT, T. L. HAYDEN, AND M. RAYDAN, Molecular conformations from distance matrices, Journal of Computational Chemistry, 14 (1993), pp. 114-120.
19. W. GLUNT, T. L. HAYDEN, AND W. M. LIU, The embedding problem for predistance matrices, Bulletin of Mathematical Biology, 53 (1991), pp. 769-796.
20. S. P. HAN, A successive projection method, Math. Programming, 40 (1988), pp. 1-14.
21. N. HIGHAM, Computing the nearest correlation matrix - a problem from finance, IMA J. Numer. Anal., 22 (2002), pp. 329-343.
22. C. HILDRETH, A quadratic programming procedure, Naval Res. Logist. Quart., 4 (1957), pp. 79-85.
23. H. HU AND I. OLKIN, A numerical procedure for finding the positive definite matrix closest to a patterned matrix, Statist. Probab. Lett., 12 (1991), pp. 511-515.
24. A. IUSEM AND A. DE PIERRO, On the convergence properties of Hildreth's quadratic programming algorithm, Math. Programming, 47 (1990), pp. 37-51.
25. A. IUSEM AND A. DE PIERRO, On the convergence of Han's method for convex programming with quadratic objective, Math. Programming, 52 (1991), pp. 265-284.
26. A. IUSEM AND B. F. SVAITER, Primal-dual row-action method for convex programming, J. Optim. Theory Appl., 86 (1995), pp. 73-112.
27. A. LENT AND Y. CENSOR, Extensions of Hildreth's row-action method for quadratic programming, SIAM J. Control Optim., 18 (1980), pp. 444-454.
28. M. MENDOZA, M. RAYDAN, AND P. TARAZAGA, Computing the nearest diagonally dominant matrix, Numer. Linear Algebra Appl., 5 (1998), pp. 461-474.
29. M. RAYDAN AND P. TARAZAGA, Primal and polar approach for computing the symmetric diagonally dominant projection, Numer. Linear Algebra Appl., 9 (2002), pp. 333-345.
30. J. VON NEUMANN, Functional operators, II. The Geometry of Orthogonal Spaces, Ann. of Math. Stud. 22, Princeton University Press, Princeton, NJ, 1950.