Convergence of a relaxed inertial proximal algorithm for maximally monotone operators
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Convergence of a relaxed inertial proximal algorithm for maximally monotone operators Hedy Attouch1 · Alexandre Cabot2 Received: 13 February 2018 / Accepted: 24 June 2019 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019
Abstract In a Hilbert space H, given A: H → 2H a maximally monotone operator, we study the convergence properties of a general class of relaxed inertial proximal algorithms. This study aims to extend to the case of the general monotone inclusion Ax 0 the acceleration techniques initially introduced by Nesterov in the case of convex minimization. The relaxed form of the proximal algorithms plays a central role. It comes naturally with the regularization of the operator A by its Yosida approximation with a variable parameter, a technique recently introduced by Attouch–Peypouquet (Math Program Ser B, 2018. https://doi.org/10.1007/s10107-018-1252-x) for a particular class of inertial proximal algorithms. Our study provides an algorithmic version of the convergence results obtained by Attouch–Cabot (J Differ Equ 264:7138–7182, 2018) in the case of continuous dynamical systems. Keywords Maximally monotone operators · Yosida regularization · Inertial proximal method · Large step proximal method · Lyapunov analysis · (Over)Relaxation Mathematics Subject Classification 49M37 · 65K05 · 65K10 · 90C25
1 Introduction Throughout this paper, H is a real Hilbert space endowed with the scalar product ., . and the corresponding norm .. Given A : H → 2H a maximally monotone operator,
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Alexandre Cabot [email protected] Hedy Attouch [email protected]
1
Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
2
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Univ. Bourgogne Franche-Comté, 21000 Dijon, France
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H. Attouch, A. Cabot
we will study the convergence properties of a general class of inertial proximal based algorithms that aim to solve the inclusion Ax 0, whose solution set is denoted by zer A. Given initial data x0 , x1 ∈ H, we consider the Relaxed Inertial Proximal Algorithm, (RIPA) for short, defined by, for k ≥ 1 (RIPA)
yk = xk + αk (xk − xk−1 ) xk+1 = (1 − ρk )yk + ρk Jμk A (yk ).
In the above formula, JμA = (I + μA)−1 is the resolvent of A with index μ > 0, where I is the identity operator. It plays a central role in the analysis of (RIPA), along with the Yosida regularization of A with parameter μ > 0, which is defined by Aμ = μ1 I − JμA , (see “Appendix A” for their main properties). We assume the following set of hypotheses ⎧ ⎨ • A : H → 2H is a maximally monotone operator; • (αk ) is a sequence of nonnegative numbers; ⎩ • (μk ) and (ρk ) are sequences of positive numbers.
(H )
If ρk = 1 for every k ≥ 1, then algorithm (RIPA) reduces to the Inertial Proximal Algorithm (IPA)
yk = xk + αk (xk − xk−1 ) xk+1 = Jμk A (yk ).
On the other hand, if αk = 0 for every k ≥ 1, then algorithm (RIPA) boils
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