On the forward–backward method with nonmonotone linesearch for infinite-dimensional nonsmooth nonconvex problems.
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| Title: | On the forward–backward method with nonmonotone linesearch for infinite-dimensional nonsmooth nonconvex problems. |
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| Authors: | Azmi, Behzad1 (AUTHOR) behzad.azmi@uni-konstanz.de, Bernreuther, Marco2 (AUTHOR) marco.bernreuther@iff.uni-stuttgart.de |
| Source: | Computational Optimization & Applications. Jul2025, Vol. 91 Issue 3, p1263-1308. 46p. |
| Subjects: | Computational mathematics, Applied mathematics, Partial differential equations, Nonlinear differential equations, Eigenfunctions |
| 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| 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. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 09266003 |
| DOI: | 10.1007/s10589-025-00684-x |
