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Approximate iterations in Bregman-function-based proximal algorithms. (English) Zbl 0920.90117
Summary: This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately. The problem being solved is modeled as a general maximal monotone operator, and need not reduce to minimization of a function. The accuracy conditions on the iterates resemble those required for the classical “linear” proximal point algorithm, but are slightly stronger; they should be easier to verify or enforce in practice than conditions given in earlier analyses of approximate generalized proximal methods. Subjects to these practically enforceable accuracy restrictions, convergence is obtained under the same conditions currently established for exact Bregman-function-based proximal methods.

90C25Convex programming
Full Text: DOI
[1] H. H. Bauschke, J. M. Borwein, Legendre functions and the method of random Bregman projections, Journal of Convex Analysis 4 (1997) 27--67. · Zbl 0894.49019
[2] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groups de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. · Zbl 0252.47055
[3] R. E. Bruck, An iterative solution of a variational inequality for certain monotone operators in Hilbert space, Bulletin of the American Mathematical Society 81 (1975) 890--892. · Zbl 0332.49005 · doi:10.1090/S0002-9904-1975-13874-2
[4] R. E. Bruck, Corrigendum to [3], Bulletin of the American Mathematical Society 82 (1976) 353.
[5] R. S. Burachik, A. N. Iusem, A generalized proximal point algorithm for the variational inequality problem in Hilbert space, Working Paper, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1995.
[6] R. S. Burachik, A. N. Iusem, B. F. Svaiter, Enlargement of monotone operators with applications to variational inequalities, Set-Valued Analysis 5 (1997) 159--180. · Zbl 0882.90105 · doi:10.1023/A:1008615624787
[7] Y. Censor, S. A. Zenios, The proximal minimization algorithm withD-functions, Journal of Optimization Theory and Applications 73 (1992) 451--464. · Zbl 0794.90058 · doi:10.1007/BF00940051
[8] Y. Censor, A. N. Iusem, S. A. Zenios, An interior-point method with Bregman functions for the variational inequality problem with paramonotone operators, Working Paper, University of Haifa, 1994. · Zbl 0919.90123
[9] G. Chen, M. Teboulle, A convergence analysis of proximal-like minimization algorithms using Bregman functions, SIAM Journal on Optimization 3 (1993) 538--543. · Zbl 0808.90103 · doi:10.1137/0803026
[10] G. Cohen, Auxiliary problem principle and decomposition of optimization problems, Journal of Optimization Theory and Applications 32 (1980) 277--305. · Zbl 0417.49046 · doi:10.1007/BF00934554
[11] G. Cohen, Auxiliary problem principle extended to variational inequalities, Journal of Optimization Theory and Applications 59 (1988) 325--333. · Zbl 0628.90069 · doi:10.1007/BF00940305
[12] J. Eckstein, Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming, Mathematics of Operations Research 18 (1993) 202--226. · Zbl 0807.47036 · doi:10.1287/moor.18.1.202
[13] J. Eckstein, M. C. Ferris, Smooth methods of multipliers for monotone complementarity problems, RUTCOR Research Report RRR 27--96, Rutgers University, 1996, revised 1997.
[14] S. D. Fl{”m, A. S. Antipin, Equilibrium programming using a proximal-like algorithm, Mathematical Programming 78 (1997) 29--41. · Zbl 0890.90150 · doi:10.1007/BF02614504
[15] A. N. Iusem, B. F. Svaiter, M. Teboulle, Entropy-like proximal methods in convex programming, Mathematics of Operations Research 19 (1994) 790--814. · Zbl 0821.90092 · doi:10.1287/moor.19.4.790
[16] A. N. Iusem, M. Teboulle, Convergence rate analysis of nonquadratic proximal methods for convex and linear programming, Mathematics of Operations Research 20 (1995) 657--677. · Zbl 0845.90099 · doi:10.1287/moor.20.3.657
[17] S. Kabbadj, Methodes Proximales Entopiques, Doctoral Thesis, Université de Montpellier II--Sciences et Techniques du Languedoc, 1994.
[18] K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM Journal on Control and Optimization 35 (1997) 1142--1168. · Zbl 0890.65061 · doi:10.1137/S0363012995281742
[19] K. C. Kiwiel, On the twice deifferentiable cubic augmented lagrangian, Journal of Optimization Theory and Applications 88 (1996) 233--236. · Zbl 0842.90093 · doi:10.1007/BF02192031
[20] S. S. Nielsen, S. A. Zenios, Proximal minimization withD-functions and massively parallel solution of linear stochastic network problems, Computational Optimization and Applications 1 (1993) 375--398. · Zbl 0784.90081 · doi:10.1007/BF00248763
[21] B. Polyak, Introduction to Optimization, Optomization Softwave Inc., New York, 1987. · Zbl 0708.90083
[22] R. T., Rockafellar, Local boundedness of nonlinear monotone operators, Michigan Mathematical Journal 16 (1969) 397--407. · Zbl 0184.17801 · doi:10.1307/mmj/1029000324
[23] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. · Zbl 0193.18401
[24] R. T. Rockafellar, On the miximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society 149 (1970) 75--88. · Zbl 0222.47017 · doi:10.1090/S0002-9947-1970-0282272-5
[25] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization 14 (1976) 877--898. · Zbl 0358.90053 · doi:10.1137/0314056
[26] M. Teboulle, Entropic proximal mappings with applications to nonlinear programming, Mathematics of Operations Research 17 (1992) 670--690. · Zbl 0766.90071 · doi:10.1287/moor.17.3.670
[27] M. Teboulle, Convergence of proximal-like algorithms, SIAM Journal of Optimization (to appear). · Zbl 0890.90151