Abstract
In this paper, we investigate the strong convergence of an inexact proximal-point algorithm. It is known that the proximal-point algorithm converges weakly to a solution of a maximal monotone operator, but fails to converge strongly. Solodov and Svaiter (Math. Program. 87:189–202, 2000) introduced a new proximal-type algorithm to generate a strongly convergent sequence and established a convergence result in Hilbert space. Subsequently, Kamimura and Takahashi (SIAM J. Optim. 13:938–945, 2003) extended the Solodov and Svaiter result to the setting of uniformly convex and uniformly smooth Banach space. On the other hand, Rockafellar (SIAM J. Control Optim. 14:877–898, 1976) gave an inexact proximal-point algorithm which is more practical than the exact one. Our purpose is to extend the Kamimura and Takahashi result to a new inexact proximal-type algorithm. Moreover, this result is applied to the problem of finding the minimizer of a convex function on a uniformly convex and uniformly smooth Banach space.
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Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal-point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000)
Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2003)
Rockafellar, R.T.: Monotone operators and the proximal-point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Martinet, B.: Regularisation d’inequations variationnelles par approximations successives. Rev. Francaise d’Automatique Inform. Rech. Opér. 4, 154–159 (1970)
Brézis, H., Lions, P.L.: Produits infinis de resolvants. Israel J. Math. 29, 329–345 (1978)
Lions, P.L.: Une methode iterative de resolution d’une inequation variationnelle. Israel J. Math. 31, 204–208 (1978)
Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)
Guler, O.: On the convergence of the proximal-point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)
Solodov, M.V., Svaiter, B.F.: A hybrid projection proximal-point algorithm. J. Convex Anal. 6, 59–70 (1999)
Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226–240 (2000)
Kamimura, S., Takahashi, W.: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal. 8, 361–374 (2000)
Eckstein, J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. Program. 83, 113–123 (1998)
Chen, G.Y., Teboulle, M.: A proximal-based decomposition method for convex minimization problem. Math. Program. 64, 81–101 (1994)
Han, D.R., He, B.S.: A new accuracy criterion for approximate proximal-point algorithms. J. Math. Anal. Appl. 263, 343–354 (2001)
Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)
Jung, J.S., Morales, C.H.: The Mann process for perturbed m-accretive operators in Banach spaces. Nonlinear Anal. 46, 231–243 (2001)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Rockafellar, R.T.: Characterization of the subdifferentials of convex functions. Pac. J. Math. 149, 75–88 (1970)
Xu, Z.B., Roach, G.F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189–210 (1991)
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Communicated by S. Schaible.
L.C. Zeng’s research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in Shanghai.
J.C. Yao’s research was partially supported by the National Science Council of the Republic of China.
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Zeng, L.C., Yao, J.C. An Inexact Proximal-Type Algorithm in Banach Spaces. J Optim Theory Appl 135, 145–161 (2007). https://doi.org/10.1007/s10957-007-9261-6
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DOI: https://doi.org/10.1007/s10957-007-9261-6