# First-Order Primal–Dual Methods for Nonsmooth Non-convex Optimisation

@article{Valkonen2019FirstOrderPM,
title={First-Order Primal–Dual Methods for Nonsmooth Non-convex Optimisation},
author={Tuomo Valkonen},
journal={arXiv: Optimization and Control},
year={2019}
}
• T. Valkonen
• Published 30 September 2019
• Mathematics
• arXiv: Optimization and Control
We provide an overview of primal-dual algorithms for nonsmooth and non-convex-concave saddle-point problems. This flows around a new analysis of such methods, using Bregman divergences to formulate simplified conditions for convergence.
4 Citations
Multikernel Regression with Sparsity Constraint | SIAM Journal on Mathematics of Data Science | Vol. 3, No. 1 | Society for Industrial and Applied Mathematics
• Computer Science, Mathematics
• 2021
A Banach-space formulation of supervised learning with generalized totalvariation (gTV) regularization is provided, and it is shown that the solution admits a multikernel expansion with adaptive positions.
Multikernel Regression with Sparsity Constraint
• Mathematics, Computer Science
SIAM J. Math. Data Sci.
• 2021
This paper identifies the class of kernel functions that are admissible in this Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization and proposes a variation of supervisedLearning in a continuous-domain hybrid search space with gTV regularization.
Regularisation, optimisation, subregularity
Regularisation theory in Banach spaces, and non-norm-squared regularisation even in finite dimensions, generally relies upon Bregman divergences to replace norm convergence. This is comparable to the
Primal-dual block-proximal splitting for a class of non-convex problems
• Mathematics
ArXiv
• 2019
We develop block structure adapted primal-dual algorithms for non-convex non-smooth optimisation problems whose objectives can be written as compositions $G(x)+F(K(x))$ of non-smooth block-separable

## References

SHOWING 1-10 OF 91 REFERENCES
and T
• Valkonen, PrimalâĂŞdual proximal splitting and generalized conjugation in non-smooth non-convex optimization,
• 2019
Primal-dual proximal splitting and generalized conjugation in nonsmooth nonconvex optimization, Applied Mathematics and Optimization (2020), doi:10.1007/s00245-020-09676-1, arXiv:1901.02746
• 1901
Primal-dual block-proximal splitting for a class of non-convex problems
• Mathematics
ArXiv
• 2019
We develop block structure adapted primal-dual algorithms for non-convex non-smooth optimisation problems whose objectives can be written as compositions $G(x)+F(K(x))$ of non-smooth block-separable
Testing and Non-linear Preconditioning of the Proximal Point Method
• T. Valkonen
• Computer Science
Applied Mathematics & Optimization
• 2018
This work formalises common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality and demonstrates the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants.
and T
• Valkonen, Primal–dual block-proximal splitting for a class of non-convex problems,
• 2019
Acceleration and Global Convergence of a First-Order Primal-Dual Method for Nonconvex Problems
• Mathematics
SIAM J. Optim.
• 2019
The primal-dual hybrid gradient method, modified (PDHGM, also known as the Chambolle--Pock method), has proved very successful for convex optimization problems involving linear operators arising in...
A primal-dual hybrid gradient method for non-linear operators with applications to MRI
We study the solution of minimax problems $\min_x \max_y G(x) + \langle K(x),y\rangle - F^*(y)$ in finite-dimensional Hilbert spaces. The functionals $G$ and $F^*$ we assume to be convex, but the
and a at
• Chemistry
The William Makepeace Thackeray Library
• 2018
The xishacorene natural products are structurally unique apolar diterpenoids that feature a bicyclo[3.3.1] framework. These secondary metabolites likely arise from the well-studied, structurally
Relaxed Gauss-Newton methods with applications to electrical impedance tomography
• Mathematics
SIAM J. Imaging Sci.
• 2020
It is proved that the Gauss--Newton-type method with inexact relaxed steps converges to a set of disjoint critical points given that the linearisation of the forward operator for the inverse problem is sufficiently precise.
Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds
• Mathematics
Found. Comput. Math.
• 2021
This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the