This paper considers the problem of designing accelerated gradient-based algorithms for optimization and saddle-point problems. The class of objective functions is defined by a generalized sector condition. This class of functions contains strongly convex functions with Lipschitz gradients but also non-convex functions, which allows not only to address optimization problems but also saddle-point problems. The proposed design procedure relies on a suitable class of Lyapunov functions and on convex semi-definite programming. The proposed synthesis allows the design of algorithms that reach the performance of state-of-the-art accelerated gradient methods and beyond.
%0 Journal Article
%1 GraEbe21
%A Gramlich, Dennis
%A Ebenbauer, Christian
%A Scherer, Carsten W.
%D 2021
%J Syst. Control Lett. (accepted)
%K EXC2075 PN4 curated
%T Convex Synthesis of Accelerated Gradient Algorithms for Optimization and Saddle Point Problems using Lyapunov functions
%U https://arxiv.org/abs/2006.09946
%X This paper considers the problem of designing accelerated gradient-based algorithms for optimization and saddle-point problems. The class of objective functions is defined by a generalized sector condition. This class of functions contains strongly convex functions with Lipschitz gradients but also non-convex functions, which allows not only to address optimization problems but also saddle-point problems. The proposed design procedure relies on a suitable class of Lyapunov functions and on convex semi-definite programming. The proposed synthesis allows the design of algorithms that reach the performance of state-of-the-art accelerated gradient methods and beyond.
@article{GraEbe21,
abstract = {This paper considers the problem of designing accelerated gradient-based algorithms for optimization and saddle-point problems. The class of objective functions is defined by a generalized sector condition. This class of functions contains strongly convex functions with Lipschitz gradients but also non-convex functions, which allows not only to address optimization problems but also saddle-point problems. The proposed design procedure relies on a suitable class of Lyapunov functions and on convex semi-definite programming. The proposed synthesis allows the design of algorithms that reach the performance of state-of-the-art accelerated gradient methods and beyond.},
added-at = {2023-11-01T19:05:18.000+0100},
author = {Gramlich, Dennis and Ebenbauer, Christian and Scherer, Carsten W.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/24a0b1b71b0aff6f1830d91606f32617a/simtech},
eprint = {2006.09946},
eprintclass = {math.OC},
eprinttype = {arXiv},
file = {:GraEbe20 - Convex Synthesis of Accelerated Gradient Algorithms for Optimization and Saddle Point Problems Using Lyapunov Functions.pdf:PDF},
interhash = {cb1163a79ce938d1cb7a17b9cdef96fa},
intrahash = {4a0b1b71b0aff6f1830d91606f32617a},
journal = {Syst. Control Lett. (accepted)},
keywords = {EXC2075 PN4 curated},
timestamp = {2023-11-08T16:37:05.000+0100},
title = {Convex Synthesis of Accelerated Gradient Algorithms for Optimization and Saddle Point Problems using Lyapunov functions},
url = {https://arxiv.org/abs/2006.09946},
year = 2021
}
