# Introduction to Nonsmooth Analysis and Optimization

@article{Clason2020IntroductionTN, title={Introduction to Nonsmooth Analysis and Optimization}, author={Christian Clason and Tuomo Valkonen}, journal={arXiv: Optimization and Control}, year={2020} }

These notes aim to give an introduction to generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for infinite-dimensional nondifferentiable optimization problems that arise in inverse problems, imaging, and PDE-constrained optimization. They cover convex subdifferentials, Fenchel duality, monotone operators and resolvents, Moreau--Yosida regularization as well as Clarke and (briefly) limiting subdifferentials. Both first-order (proximal…

## 17 Citations

### Regularisation, optimisation, subregularity

- MathematicsInverse Problems
- 2020

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…

### Proximal methods for point source localisation

- MathematicsArXiv
- 2022

Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed…

### A nonsmooth primal-dual method with simultaneous adaptive PDE constraint solver

- MathematicsArXiv
- 2022

We introduce an ecient rst-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this eciency through not solving the PDE or its linearisation on…

### A Newton derivative scheme for shape optimization problems constrained by variational inequalities

- Mathematics, Computer Science
- 2022

This paper considers Newton-derivatives in order to formulate optimality conditions for shape optimization problems constrained by variational inequalities and sets up a Newton-shape derivative scheme.

### Regularization for Wasserstein Distributionally Robust Optimization

- Computer Science, Mathematics
- 2022

This paper derives a general strong duality result of regularized Wasserstein distributionally robust problems in the case of entropic regularization and provides an approximation result when the regularization parameters vanish.

### Linearly convergent bilevel optimization with single-step inner methods

- Mathematics
- 2022

We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an…

### bilevel optimization with single-step inner methods

- Mathematics
- 2022

We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an…

### Optimal regularized hypothesis testing in statistical inverse problems

- MathematicsArXiv
- 2022

Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of Inverse Problems, where the quantity of interest is not…

### Numerical approximation of regularized non--convex elliptic optimal control problems by the finite element method

- Mathematics, Computer Science
- 2022

. We investigate the numerical approximation of an elliptic optimal control problem which involves a nonconvex local regularization of the L q -quasinorm penalization (with q ∈ (0 , 1)) in the cost…

### Regularizing orientation estimation in Cryo-EM 3D map refinement through measure-based lifting over Riemannian manifolds

- Business
- 2022

Motivated by the trade-off between noise-robustness and data-consistency for joint 3D map reconstruction and rotation estimation in single particle cryogenic-electron microscopy (Cryo-EM), we propose…

## References

SHOWING 1-10 OF 147 REFERENCES

### Variational analysis and generalized differentiation

- Economics
- 2006

Applications.- Constrained Optimization and Equilibria.- Optimal Control of Evolution Systems in Banach Spaces.- Optimal Control of Distributed Systems.- Applications to Economics.

### Variational analysis and applications, Springer monographs in mathematics

- MathematicsOptimization
- 2019

### Functional Analysis, Calculus of Variations and Optimal Control

- Mathematics
- 2013

Normed Spaces.- Convex sets and functions.- Weak topologies.- Convex analysis.- Banach spaces.- Lebesgue spaces.- Hilbert spaces.- Additional exercises for Part I.- Optimization and multipliers.-…

### Sur quelques points de méthodologie géométrique

- Rev. Gén. des Sciences
- 1930

### Nonsmooth Analysis, Universitext, Springer, Berlin, doi: 10.1007/9783-540-71333-3

- 2007

### Functional Analysis, 2nd ed., McGraw-Hill, New York

- 1991

### First-Order Methods in Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, doi: 10.1137/1.9781611974997

- 2017

### Primal-dual block-proximal splitting for a class of non-convex problems

- MathematicsArXiv
- 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…