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This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz…
In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool,…
This paper investigates a specific class of nonsmooth nonconvex optimization problems in the face of data uncertainty, namely, robust optimization problems, where the given objective function can be expressed as a difference of two…
Nonconvex functionals with spherical symmetry are studied. Existence of one and radial symmetry of all global minimizers is shown with an approach based on convex relaxation.
In this paper we consider a class of non-local in time telegraph equations. Recently, it has been proved that the fundamental solutions of such equations can be interpreted as the probability density function of a stochastic process. We…
This paper investigates the application of mini-batch gradient descent to semiflows (gradient flows). Given a loss function (potential), we introduce a continuous version of mini-batch gradient descent by randomly selecting sub-loss…
The article is devoted to the investigation of smoothness of functions $f(x_1,...,x_m)$ of variables $x_1,...,x_m$ in infinite fields with non-trivial multiplicative ultra-norms, where $m\ge 2$. Theorems about classes of smoothness $C^n$ or…
We introduce a new notion of Almgren's frequency which is adapted to solutions of a sub-Laplacian (harmonic functions) on a Carnot group of arbitrary step $\bG$. With this notion we investigate some new functionals associated with the…
We prove convergence of a single time-scale stochastic subgradient method with subgradient averaging for constrained problems with a nonsmooth and nonconvex objective function having the property of generalized differentiability. As a tool…
In this paper we propose a generalized condition for a sharp minimum, somewhat similar to the inexact oracle proposed recently by Devolder-Glineur-Nesterov. The proposed approach makes it possible to extend the class of applicability of…
Submodular functions, defined on continuous or discrete domains, arise in numerous applications. We study the minimization of the difference of two submodular (DS) functions, over both domains, extending prior work restricted to set…
This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional…
We study the set of continuous functions that admit no spurious local optima (i.e. local minima that are not global minima) which we term \textit{global functions}. They satisfy various powerful properties for analyzing nonconvex and…
We prove some basic properties of quasinearly subharmonic functions and quasinearly subharmonic functions in the narrow sense.
Stochastic methods for minimizing a convex integral functional, as initiated by Robbins and Monro in the early 1950s, rely on the evaluation of a gradient (or subgradient if the function is not smooth) and moving in the corresponding…
In this paper, we first study nonsmooth steepest descent method for nonsmooth functions defined on Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for…
We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the…
Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fr\'echet subdifferentials in Asplund spaces.
We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a…
The purpose of this article is to present a result on the existence of Cauchy temporal functions invariant by the action of a compact group of conformal transformations in arbitrary globally hyperbolic manifolds. Moreover, the previous…