Abstract
We propose an inexact version of the proximal point method and study its properties in nonreflexive Banach spaces which are duals of separable Banach spaces, both for the problem of minimizing convex functions and of finding zeroes of maximal monotone operators. By using surjectivity results for enlargements of maximal monotone operators, we prove existence of the iterates in both cases. Then we recover most of the convergence properties known to hold in reflexive and smooth Banach spaces for the convex optimization problem. When dealing with zeroes of monotone operators, our convergence result requests that the regularization parameters go to zero, as is the case for standard (non-proximal) regularization schemes.
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References
Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. Reidel, Dordrecht (1986)
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 Comp. Math. Math. Phys. 7, 200–217 (1967)
Burachik, R.S., Iusem, A.N.: Set-valued Mappings and Enlargements of Monotone Operators. Springer, New York (2007)
Burachik, R.S., Scheimberg, S.: A proximal point algorithm for the variational inequality problem in Banach spaces. SIAM J. Control Optim. 39, 1615–1632 (2001)
Butnariu D., Iusem, A.N.: On a proximal point method for convex optimization in Banach spaces. Numer. Funct. Anal. Optim. 18, 723–744 (1997)
Censor, Y., Iusem, A.N., Zenios, S.A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Program. 81, 373–400 (1998)
Gossez, J.P.: Opérateurs monotones nonlineaires dans les espaces de Banach nonreflexifs. J. Math. Anal. Appl. 34, 371–395 (1971)
Hanke, M., Groetsch, C.W.: Nonstationary iterated Tikhonov regularization. J. Optim. Theory Appl. 98, 37–53 (1998)
He, L., Burger, M., Osher, S.: Iterative total variation regularization with non-quadratic fidelity. J. Math. Imaging Vis. 26, 167–184 (2006)
Iusem, A.N., Gárciga, R.O.: Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer. Funct. Anal. Optim. 22, 609–640 (2001)
Kassay, G.: The proximal points algorithm for reflexive Banach spaces. Studia Univ. Babes-Bolyai, Mathematica 30, 9–17 (1985)
Marques Alves, M., Svaiter, B.: On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces. JNA (2008, submitted). arXiv:0805.4609v1
Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Francaise Inform. Rech. Oper. 3, 154–179 (1970)
Meyer, Y.: Oscillatory Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22. American Mathematical Society, Providence (2001)
Moreau, J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. math. Fr. 93, 273–299 (1965)
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation based image restoration. Multiscale Modelling and Simulation 4, 460–489 (2005)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm in convex programming. SIAM J. Control Optim. 14, 877–898 (1976)
Simons, S.: The range of a monotone operator. J. Math. Anal. Appl. 199, 176–201 (1996)
Tikhonov, A., Arsenin, V.: Solution of Ill-posed Problems. Wiley, New York (1977)
Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey (2002)
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Iusem, A.N., Resmerita, E. A Proximal Point Method in Nonreflexive Banach Spaces. Set-Valued Anal 18, 109–120 (2010). https://doi.org/10.1007/s11228-009-0126-z
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DOI: https://doi.org/10.1007/s11228-009-0126-z