On the Linear Convergence of Forward–Backward Splitting Method: Part I—Convergence Analysis

Yunier Bello-Cruz (), Guoyin Li () and Tran T. A. Nghia ()
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Yunier Bello-Cruz: Northern Illinois University
Guoyin Li: University of New South Wales
Tran T. A. Nghia: Oakland University

Journal of Optimization Theory and Applications, 2021, vol. 188, issue 2, No 4, 378-401

Abstract: Abstract In this paper, we study the complexity of the forward–backward splitting method with Beck–Teboulle’s line search for solving convex optimization problems, where the objective function can be split into the sum of a differentiable function and a nonsmooth function. We show that the method converges weakly to an optimal solution in Hilbert spaces, under mild standing assumptions without the global Lipschitz continuity of the gradient of the differentiable function involved. Our standing assumptions is weaker than the corresponding conditions in the paper of Salzo (SIAM J Optim 27:2153–2181, 2017). The conventional complexity of sublinear convergence for the functional value is also obtained under the local Lipschitz continuity of the gradient of the differentiable function. Our main results are about the linear convergence of this method (in the quotient type), in terms of both the function value sequence and the iterative sequence, under only the quadratic growth condition. Our proof technique is direct from the quadratic growth conditions and some properties of the forward–backward splitting method without using error bounds or Kurdya-Łojasiewicz inequality as in other publications in this direction.

Keywords: Nonsmooth and convex optimization problems; Forward–Backward splitting method; Linear convergence; Quadratic growth condition; 65K05; 90C25; 90C30 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10957-020-01787-7

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