• Corpus ID: 209532137

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… 

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References

SHOWING 1-10 OF 147 REFERENCES

Variational analysis and generalized differentiation

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

Functional Analysis, Calculus of Variations and Optimal Control

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

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
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