An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions

References

• Bauschke H. H., Borwein J. M. Legendre functions and the method of random Bregman projections. J. Convex Anal. (1997) 4:27–67Google Scholar
• Bregman L. M. The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. and Math. Phys. (1967) 7(3):200–217Crossref, Google Scholar
• Brézis H., Lions P. L. Produits infinis de résolvantes. Israel J. Math. (1978) 29:329–345Crossref, Google Scholar
• Bruck R. E. An iterative solution of a variational inequality for certain monotone operators in a Hilbert space. Bull. Amer. Math. Soc. (1975) 81:890–892Crossref, Google Scholar
• Bruck R. E. Corrigendum to [4]. Bull. Amer. Math. Soc. (1976) 82:353Google Scholar
• Burachik R. S., Iusem A. N. A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM J. Optim. (1998) 8:197–216Crossref, Google Scholar
• Burachik R. S., Iusem A. N., Svaiter B. F. Enlargement of monotone operators with applications to variational inequalities. Set-Valued Anal. (1997) 5:159–180Crossref, Google Scholar
• Burke J. V., Qian M. A variable metric proximal point algorithm for monotone operators. SIAM J. Control Optim. (1998) 37:353–375Crossref, Google Scholar
• Censor Y., Iusem A. N., Zenios S. A. An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Programming (1998) 81:373–400Crossref, Google Scholar
• Censor Y., Zenios S. The proximal minimization algorithm with D-functions. J. Optim. Theory Appl. (1992) 73:451–464Crossref, Google Scholar
• Chen G., Teboulle M. Convergence analysis of proximal-like optimization algorithm using Bregman functions. SIAM J. Optim. (1993) 3:538–543Crossref, Google Scholar
• Cottle R. W., Giannessi F., Lions J.-L.Variational Inequalities and Complementarity Problems: Theory and Applications (1980) (Wiley, New York) Google Scholar
• Eckstein J. Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Math. Oper. Res. (1993) 18:202–226Link, Google Scholar
• Eckstein J. Approximate iterations in Bregman-function-based proximal algorithms. Math. Programming (1998) 83:113–123Crossref, Google Scholar
• Eckstein J., Bertsekas D. P. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathe. Programming (1992) 55:293–318Crossref, Google Scholar
• Eggermont P. P. B. Multiplicative iterative algorithms for convex programming. Linear Algebra Appl. (1990) 130:25–32Crossref, Google Scholar
• Ferris M. C. Finite termination of the proximal point algorithm. Math. Programming (1991) 50:359–366Crossref, Google Scholar
• Güler O. New proximal point algorithms for convex minimization. SIAM J. Optim. (1992) 2:649–664Crossref, Google Scholar
• Iusem A. N. On some properties of generalized proximal point methods for quadratic and linear programming. J. Optim. Theory Appl. (1995) 85:596–612Crossref, Google Scholar
• Iusem A. N. On some properties of generalized proximal point methods for the variational inequality problem. J. Optim. Theory Appl. (1998a) 96:337–362Crossref, Google Scholar
• Iusem A. N. On some properties of paramonotone operators. J. Convex Anal. (1998b) 5:269–278Google Scholar
• Iusem A. N., Solodov M. V. Newton-type methods with generalized distances for constrained optimization. Optim. (1997) 41:257–278Crossref, Google Scholar
• Kiwiel K. C. Proximal minimization methods with generalized Bregman functions. SIAM J. Control Optim. (1997) 35:1142–1168Crossref, Google Scholar
• Lemaire B., Penot J. P. The proximal algorithm. New Methods of Optimization and Their Industrial Use (1989) 87(Birkhauser, Basel, Switzerland) 73–87International Series of Numerical MathematicsGoogle Scholar
• Luque F. J. Asymptotic convergence analysis of the proximal point algorithm. SIAM J. Control Optim. (1984) 22:277–293Crossref, Google Scholar
• Martinet B. Regularisation d'inequations variationelles par approximations successives. Rev. Française Inform. Rech. Opér. (1970) 4:154–159Google Scholar
• Moreau J.-J. Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. France (1965) 93:273–299Crossref, Google Scholar
• Pang J.-S., Horst R., Pardalos P. Complementarity problems. Handbook of Global Optimization (1995) (Kluwer Academic Publishers, Boston, MA) 271–338Crossref, Google Scholar
• Passty G. B. Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. (1979) 67:274–276Crossref, Google Scholar
• Rockafellar R. T. Local boundedness of nonlinear monotone operators. Michigan Math. J. (1969) 16:397–407Crossref, Google Scholar
• Rockafellar R. T. On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. (1970a) 149:75–88Crossref, Google Scholar
• Rockafellar R. T.Convex Analysis (1970b) (Princeton University Press, Princeton, NJ) Crossref, Google Scholar
• Rockafellar R. T. Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. (1976a) 1:97–116Link, Google Scholar
• Rockafellar R. T. Monotone operators and the proximal point algorithm. SIAM J. Control Optim. (1976b) 14:877–898Crossref, Google Scholar
• Solodov M. V., Svaiter B. F. A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem. SIAM J. Optim. (2000a) 10:605–625Crossref, Google Scholar
• Solodov M. V., Svaiter B. F., Di Pillo G., Giannessi F. A comparison of rates of convergence of two inexact proximal point algorithms. Nonlinear Optimization and Related Topics (2000b) (Kluwer Academic Publishers). ForthcomingCrossref, Google Scholar
• Solodov M. V., Svaiter B. F. Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Programming (2000c) 87:189–202Crossref, Google Scholar
• Solodov M. V., Svaiter B. F., Fukushima M., Qi L. A globally convergent inexact Newton method for systems of monotone equations. Reformulation—Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (1999a) (Kluwer Academic Publishers, Dordrecht, The Netherlands) 355–369Google Scholar
• Solodov M. V., Svaiter B. F. A hybrid approximate extragradient—proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Anal. (1999c) 7:323–345Crossref, Google Scholar
• Solodov M. V., Svaiter B. F. A hybrid projection—proximal point algorithm. J. Convex Anal. (1999b) 6(1):59–70Google Scholar
• Teboulle M. Convergence of proximal-like algorithms. SIAM J. Optim. (1997) 7:1069–1083Crossref, Google Scholar