Related papers: Gradient flows of $(K,N)$-convex functions with ne…

We investigate the computation of the gradient of the value function in parametric convex optimization problems. We derive general expression for the gradient of the value function in terms of the cost function, constraints and Lagrange…

We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are…

A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schr\"odinger equations.…

Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…

We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are…

In this work we give a proof of the mean-field limit for $\lambda$-convex potentials using a purely variational viewpoint. Our approach is based on the observation that all evolution equations that we study can be written as gradient flows…

In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a $n$-dimensional convex domain, and show a weak continuity theorem with respect to pointwise…

We disclose an interesting connection between 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 $\nabla\psi$, under adequate convexity…

We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Gamma-converging functionals and the gradient flow motion for the corresponding limit functional, in a…

We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained…

For large classes of non-convex subsets $Y$ in ${\mathbb R}^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$…

It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as…

In this paper we give a characterization of all order isomorphisms on some classes of convex functions. We deal with the class $Cvx(K)$ consisting of lower-semi-continuous convex functions defined on a convex set $K$, and its subclass…

We study geometric properties of the gradient flow for learning deep linear convolutional networks. For linear fully connected networks, it has been shown recently that the corresponding gradient flow on parameter space can be written as a…

The paper is devoted to the study, characterizations, and applications of variational convexity of functions, the property that has been recently introduced by Rockafellar together with its strong counterpart. First we show that these…

We show that convexity of the effective action follows from its functional flow equation. Our analysis is based on a new, spectral representation. The results are relevant for the study of physical instabilities. We also derive constraints…

This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple geometric property. Then we generalize E-convex…

The symplectic vortex equations admit a variational description as global minimum of the Yang-Mills-Higgs functional. We study its negative gradient flow on holomorphic pairs $(A,u)$ where $A$ is a connection on a principal $G$-bundle $P$…

We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy…

We analyze gradient flows with jumps generated by a finite set of complete vector fields in involution using some Radon measures $u\in \mathcal{U}_a$ as admissible perturbations. Both the evolution of a bounded gradient flow $\{x^u(t,\l)\in…