On the convergence rate of the over-relaxed proximal point algorithm

نویسندگان

  • Zhenyu Huang
  • Muhammad Aslam Noor
چکیده

This paper is to illustrate that the main result of the paper [R.U. Verma, Generalized overrelaxedproximal algorithmbased onA-maximalmonotonicity framework and applications to inclusion problems, Mathematical and Computer Modelling 49 (2009) 1587–1594] is incorrect. The convergence rate of the over-relaxed proximal point algorithm should be greater than 1. Moreover, the strong convergence and the unique solution may not be proved accordingly in the paper by Verma. © 2012 Elsevier Ltd. All rights reserved. The purpose of this paper is to illustrate that the main result of [1] is incorrect. Let X be a real Hilbert space with the norm ∥ · ∥ and the inner product ⟨·, ·⟩. The class of nonlinear set-valued variational inclusions is to find a solution to 0 ∈ M(x), (1) whereM : X → 2X is a set-valued mapping on X . Definition 1. Let A : X → X be a single-valued mapping on X . The mapping A is said to be: (i) r-strongly monotone if there exists a positive constant r such that ⟨Au − Av, u − v⟩ ≥ r∥u − v∥2, ∀u, v ∈ X . (ii) s-Lipschitz continuous if there exists a positive constant s such that ∥Au − Av∥ ≤ s∥u − v∥, ∀u, v ∈ X . Readers may refer to [1] for the definitions of m-relaxed monotone mappings, A-maximal monotone mappings, generalized resolvent operators, and H-maximal monotone mappings (see [1, Definitions 2.1–2.5]). Lemma 1 (See [1, Lemma 3.1, p. 1589]). Let X be a real Hilbert space, let A : X → X be r-stronglymonotone, and let M : X → 2X be A-maximal monotone. Then the generalized resolvent operator associated with M and defined by JM ρ,A(u) = (A + ρM) −1(u) ∀u ∈ X is 1 r−ρm -Lipschitz continuous for (r − ρm) > 0. ✩ The first author is supported by the National Natural Science Foundation of China (NSFC Grant No. 10871092), the Fundamental Research Funds for the Central University of China (Grant No. 1113020301 and Grant No. 1116020301) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD Grant). ∗ Corresponding author. Tel.: +86 25 83595706. E-mail addresses: [email protected], [email protected] (Z.Y. Huang), [email protected], [email protected] (M.A. Noor). 0893-9659/$ – see front matter© 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.02.003 2 Z.Y. Huang, M.A. Noor / Applied Mathematics Letters ( ) – Lemma 2 (See [1, Lemma 3.2, p. 1589]). Let X be a real Hilbert space, let A : X → X be r-strongly monotone and s-Lipschitz continuous, and let M : X → 2X be A-maximal monotone. Then the generalized resolvent operator associated with M and defined by JM ρ,A(u) = (A + ρM) −1(u) ∀u ∈ X satisfies ∥JM ρ,A(A(u)) − J M ρ,A(A(v))∥ ≤ s r − ρm ∥u − v∥, and ⟨JM ρ,A(A(u)) − J M ρ,A(A(v)), A(u) − A(v)⟩ ≥ (r − ρm)∥J M ρ,A(A(u)) − J M ρ,A(A(v))∥ 2, where (r − ρm) > 0. In [1], the author used Lemmas 1 and 2 to obtain the following main result on the convergence rate, which holds only when (r − ρm) > 0: Theorem 3.3 of [1, p. 1590]. Let X be a real Hilbert space, let A : X → X be r-strongly monotone and s-Lipschitz continuous, and let M : X → 2X be A-maximal monotone. For an arbitrarily chosen initial point x0, suppose that the sequence {xk} is generated by the generalized proximal point algorithm A(xk+1) = (1 − αk)A(x) + αky ∀k ≥ 0, (2) and yk satisfies ∥yk − A(JM ρk,A(A(x k)))∥ ≤ δk∥y − A(xk)∥, (3) where JM ρk,A = (A + ρkM) −1, (4) and {δk}, {αk}, {ρk} ⊆ [0, ∞) are scalar sequences. Then the sequence {xk} converges linearly to a unique solution x of (1) with the convergence rate θk =  s2 r2  1 − αk  2  1 − (r − ρkm) (r − ρkm)  −  1 − 2(r − ρkm) − s2 (r − ρkm)  αk  < 1, (5) where α2 k + 2αk(1 − αk)(r − ρkm) > 0, αk > 1, s > 1, (6)

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عنوان ژورنال:
  • Appl. Math. Lett.

دوره 25  شماره 

صفحات  -

تاریخ انتشار 2012