Related papers: Minimizers of Convex Functionals Arising in Random…

We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and $2d$-smoothness results for vector valued minimizers of possibly degenerate functionals. Our…

We establish the existence of solutions to common noise McKean-Vlasov martingale problems for coefficients with low regularity. Our approach is able to handle the key challenge posed by drift coefficients that are discontinuous with respect…

We study the minimal class of exact solutions of the Saffman-Taylor problem with zero surface tension, which contains the physical fixed points of the regularized (non-zero surface tension) problem. New fixed points are found and the basin…

This paper provides a set of sensitivity analysis and activity identification results for a class of convex functions with a strong geometric structure, that we coined "mirror-stratifiable". These functions are such that there is a…

We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…

The minimization of a nonconvex composite function can model a variety of imaging tasks. A popular class of algorithms for solving such problems are majorization-minimization techniques which iteratively approximate the composite nonconvex…

Estimation of convex functions finds broad applications in engineering and science, while convex shape constraint gives rise to numerous challenges in asymptotic performance analysis. This paper is devoted to minimax optimal estimation of…

Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed…

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…

This paper addresses the optimization problem of minimizing non-convex continuous functions, which is relevant in the context of high-dimensional machine learning applications characterized by over-parametrization. We analyze a randomized…

We consider a variational model for periodic partitions of the upper half-space into three regions, where two of them have prescribed volume and are subject to the geometrical constraint that their union is the subgraph of a function, whose…

Min-max problems have broad applications in machine learning, including learning with non-decomposable loss and learning with robustness to data distribution. Convex-concave min-max problem is an active topic of research with efficient…

We establish partial H\"older regularity for (local) generalised minimisers of variational problems involving strongly quasi-convex integrands of linear growth, where the full gradient is replaced by a first order homogeneous differential…

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid…

We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),$$ where $\beta(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of…

For a class of discrete quasi convex functions called semi-strictly quasi M$^\natural$-convex functions, we investigate fundamental issues relating to minimization, such as optimality condition by local optimality, minimizer cut property,…

We prove that certain Bellman functions of several variables are the minimal locally concave functions. This generalizes earlier results about Bellman functions of two variables.

Let F be nonnegative, convex and smooth off a compact set K. We prove that continuous local minimisers of convex functionals are "very weak" viscosity solutions in the sense of Juutinen-Lindqvist of the highly singular Euler-Lagrange PDE…

We show that minimizers of convex functions subject to almost all linear perturbations are nondegenerate. An analogous result holds more generally, for lower-C^2 functions.

We construct the first and second Chern-Ricci functions on negatively curved minimal surfaces in ${\mathbb{R}}^{3}$ using Gauss curvature and angle functions, and establish that they become harmonic functions on the minimal surfaces. We…