@article{KanzowMehlitz2022,
  author    = {Kanzow, Christian and Mehlitz, Patrick},
  title     = {Convergence properties of monotone and nonmonotone proximal gradient methods revisited},
  series = {Journal of Optimization Theory and Applications},
  volume    = {195},
  journal   = {Journal of Optimization Theory and Applications},
  number    = {2},
  issn      = {0022-3239},
  doi       = {10.1007/s10957-022-02101-3},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324351},
  pages     = {624-646},
  year      = {2022},
  abstract  = {Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis.},
  language  = {en}
}
@article{GaviraghiSchindeleAnnunziatoetal.2016,
  author    = {Gaviraghi, Beatrice and Schindele, Andreas and Annunziato, Mario and Borz{\`i}, Alfio},
  title     = {On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes},
  series = {Applied Mathematics},
  volume    = {7},
  journal   = {Applied Mathematics},
  number    = {16},
  doi       = {10.4236/am.2016.716162},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147819},
  pages     = {1978 -- 2004},
  year      = {2016},
  abstract  = {A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.},
  language  = {en}
}
@article{SchindeleBorzi2016,
  author    = {Schindele, Andreas and Borz{\`i}, Alfio},
  title     = {Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional},
  series = {Applied Mathematics},
  volume    = {7},
  journal   = {Applied Mathematics},
  number    = {9},
  doi       = {10.4236/am.2016.79086},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145850},
  pages     = {967-992},
  year      = {2016},
  abstract  = {First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.},
  language  = {en}
}