TY  - JOUR
A1  - Natemeyer, Carolin
A1  - Wachsmuth, Daniel
T1  - A proximal gradient method for control problems with non-smooth and non-convex control cost
JF  - Computational Optimization and Applications
N2  - We investigate the convergence of the proximal gradient method applied to control problems with non-smooth and non-convex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of L\(^{p}\)-type for p\in [0,1). We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin’s maximum principle and weaker than L-stationarity.
KW  - sparse control problems
KW  - proximal gradient method
KW  - non-smooth and non-convex optimization
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269069
SN  - 1573-2894
VL  - 80
IS  - 2
ER  - 
TY  - JOUR
A1  - Kanzow, Christian
A1  - Mehlitz, Patrick
T1  - Convergence properties of monotone and nonmonotone proximal gradient methods revisited
JF  - Journal of Optimization Theory and Applications
N2  - 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.
KW  - non-Lipschitz optimization
KW  - nonsmooth optimization
KW  - proximal gradient method
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324351
SN  - 0022-3239
VL  - 195
IS  - 2
ER  -