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

Author

Listed:
  • Behzad Azmi

    (University of Konstanz)

  • Marco Bernreuther

    (University of Stuttgart)

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.

Suggested Citation

  • Behzad Azmi & Marco Bernreuther, 2025. "On the forward–backward method with nonmonotone linesearch for infinite-dimensional nonsmooth nonconvex problems," Computational Optimization and Applications, Springer, vol. 91(3), pages 1263-1308, July.
    • Handle: RePEc:spr:coopap:v:91:y:2025:i:3:d:10.1007_s10589-025-00684-x
    DOI: 10.1007/s10589-025-00684-x
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