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Erscheinungsjahr: 
2025
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[Journal:] Computational Optimization and Applications [ISSN:] 1573-2894 [Volume:] 91 [Issue:] 3 [Publisher:] Springer US [Place:] New York [Year:] 2025 [Pages:] 1263-1308
Verlag: 
Springer US, New York
Zusammenfassung: 
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.
Schlagwörter: 
Nonsmooth nonconvex optimization
Forward–backward algorithm
Infinite-dimensional problems
Nonmonotone linesearch
Quadratic growth
PDE-constrained optimization
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