# Descent modulus and applications

@inproceedings{Daniilidis2022DescentMA, title={Descent modulus and applications}, author={Aris Daniilidis and Laurent Miclo and David Salas}, year={2022} }

. The norm of the gradient k∇ f ( x ) k measures the maximum descent of a real-valued smooth function f at x . For (nonsmooth) convex functions, this is expressed by the distance dist(0 , ∂f ( x )) of the subdiﬀerential to the origin, while for general real-valued functions deﬁned on metric spaces by the notion of metric slope |∇ f | ( x ). In this work we propose an axiomatic deﬁnition of descent modulus T [ f ]( x ) of a real-valued function f at every point x , deﬁned on a general (not…

## References

SHOWING 1-10 OF 30 REFERENCES

### Diffusion processes and heat kernels on metric spaces

- Mathematics
- 1998

Ž . processes X , P on any given locally compact metric space X, d tx equipped with a Radon measure m. These processes are associated with local regular Dirichlet forms which are obtained as -limits…

### Gradient Flows: In Metric Spaces and in the Space of Probability Measures

- Mathematics
- 2005

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence…

### Characterization of the subdifferentials of convex functions

- Mathematics
- 1966

Each lower semi-continuous proper convex function / on a Banach space E defines a certain multivalued mapping df from E to E* called the subdifferential of /. It is shown here that the mappings…

### Curves of Descent

- MathematicsSIAM J. Control. Optim.
- 2015

It is shown that curves of near-maximal slope of semialgebraic functions have a more classical description as solutions of subgradient dynamical systems as well as the existence theory is amplified for semial algebraic functions, prototypical nonpathological functions in nonsmooth optimization.

### Measure theory and fine properties of functions

- Mathematics
- 1992

GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems…

### Analysis and Geometry of Markov Diffusion Operators

- Mathematics
- 2013

Introduction.- Part I Markov semigroups, basics and examples: 1.Markov semigroups.- 2.Model examples.- 3.General setting.- Part II Three model functional inequalities: 4.Poincare inequalities.-…

### Gradient Flows, Second-Order Gradient Systems and Convexity

- MathematicsSIAM J. Optim.
- 2018

A surprising result is obtained regarding the gradient flow of a $\mathcal{C}^2$-smooth function $\psi$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nabla\psi$.

### Countably orderable sets and their applications in optimization

- Mathematics
- 1992

A special class of relations is common in many of optimization problems. We extract it in a general form and provide sufficient conditions for the existence of maximal points. The general results a...

### Markov Processes

- MathematicsIntroduction to Stochastic Processes and Simulation
- 2018

1 General Properties of Markov Processes 2 1.1 Discrete Time Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Classification . . . . . . . . . . . . . .…