Related papers: Generalized differentiation of piecewise linear fu…

In the article the author is studying the twice codifferentiable functions, defined by Prof. V.Ph. Demyanov, and some methods for calculating their codifferentials. At the beginning easier case is considered when a function is twice…

In this paper, we establish an a priori second-order estimate for admissible solutions satisfying a dynamic plurisubharmonic condition to equations involving sums of Hessian operators on compact Hermitian manifolds. The estimate is derived…

This paper is devoted to a systematic study and characterizations of the fundamental notions of variational and strong variational convexity for lower semicontinuous functions. While these notions have been quite recently introduced by…

A class of second-order algorithms is proposed for minimizing smooth nonconvex functions that alternates between regularized Newton and negative curvature steps in an iteration-dependent subspace. In most cases, the Hessian matrix is…

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We…

In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h-convex and we point out the results for some special…

In the Euclidean setting, the well-known Alexandrov theorem states that convex functions are twice differentiable almost everywhere. In this note, we extend this theorem to rank-one convex functions. Our approach is novel in that it draws…

Finite-sum optimization problems are ubiquitous in machine learning, and are commonly solved using first-order methods which rely on gradient computations. Recently, there has been growing interest in \emph{second-order} methods, which rely…

The paper proposes and develops new globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems. These algorithms are based on the theory and calculations of…

Using the notion of h-subdifferential, we characterize both first and second order differentiability of h-convex functions in stratified groups. We show that Aleksandrov's second order differentiability of h-convex functions is equivalent…

A piecewise linear function can be described in different forms: as an arbitrarily nested expression of $\min$- and $\max$-functions, as a difference of two convex piecewise linear functions, or as a linear combination of maxima of…

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…

We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations \begin{align*} P(x,D) = \sum_{\ell,j=1}^{n} a_{\ell,j}(x) D_{\ell} D_{j} + \sum_{\ell=1}^{n} i…

We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above…

I consider general reflection coefficients for arbitrary one-dimensional whole line differential or difference operators of order $2$. These reflection coefficients are semicontinuous functions of the operator: their absolute value can only…

A recent development in extreme value modeling uses the geometry of the dataset to perform inference on the multivariate tail. A key quantity in this inference is the gauge function, whose values define this geometry. Methodology proposed…

We prove upper and lower bounds for a variational functional for convex functions satisfying certain boundary conditions on a sector of the unit ball in two dimensions. The functional contains two terms: The full Hessian and its…

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope of their minimal (with respect to pointwise…

We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which…

Convexity is an important notion in non linear optimization theory as well as in infinite dimensional functional analysis. As will be seen below, very simple and powerful tools will be derived from elementary duality arguments (which are…