On the forward–backward method with nonmonotone linesearch for infinite-dimensional nonsmooth nonconvex problems

Behzad Azmi () and Marco Bernreuther ()
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Behzad Azmi: University of Konstanz
Marco Bernreuther: University of Stuttgart

Computational Optimization and Applications, 2025, vol. 91, issue 3, No 8, 1263-1308

Abstract: Abstract This paper provides a comprehensive study of the nonmonotone forward–backward splitting (FBS) method for solving a class of nonsmooth composite problems in Hilbert spaces. The objective function is the sum of a Fréchet differentiable (not necessarily convex) function and a proper lower semicontinuous convex (not necessarily smooth) function. These problems appear, for example, frequently in the context of optimal control of nonlinear partial differential equations (PDEs) with nonsmooth sparsity-promoting cost functionals. We discuss the convergence and complexity of FBS equipped with the nonmonotone linesearch under different conditions. In particular, R-linear convergence will be derived under quadratic growth-type conditions. We also investigate the applicability of the algorithm to problems governed by PDEs. Numerical experiments are also given that justify our theoretical findings.

Keywords: Nonsmooth nonconvex optimization; Forward–backward algorithm; Infinite-dimensional problems; Nonmonotone linesearch; Quadratic growth; PDE-constrained optimization; 90C26; 49M41; 65K05; 65K15; 49M37; 65J22 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-025-00684-x

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