相关论文: The Necessary and Sufficient Conditions for Repres…
In the article the necessary and sufficient conditions for a representation of Lipschitz function of more than two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome…
There are given conditions for represention of a function of many arguments as the difference of convex functions.
We give a necessary and sufficient condition ensuring that any function which is separately Lipschitz on two fixed compact sets is Lipschitz on their union.
We provide comparison principles for convex functions through its proximal mappings. Consequently, we prove that the norm of the proximal operator determines a convex the function up to a constant. A new characterization of Lipschitzianity…
We give a necessary and sufficient condition for a difference of convex (DC, for short) functions, defined on a locally convex space, to be Lipschitz continuous. Our criterion relies on the intersections of the "epsilon-subdifferentials of…
In this note, we establish the Lipschitz continuity of finite-dimensional globally convex functions on all given balls and global Lipschitz continuity for eligible functions of that type. The Lipschitz constants in both situations draw…
Lipschitz constants for the width and diameter functions of a convex body in $\mathbb R^n$ are found in terms of its diameter and thickness (maximum and minimum of both functions). Also, a dual approach to thickness is proposed.
This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the…
Necessary and sufficient conditions for Lipschitzness of the Lempert and Green functions are found in terms of their boundary behaviors.
The aim of this paper is to prove the exponential convergence, local and global, of Adam algorithm under precise conditions on the parameters, when the objective function lacks differentiability. More precisely, we require Lipschitz…
In this paper some adaptive mirror descent algorithms for problems of minimization convex objective functional with several convex Lipschitz (generally, non-smooth) functional constraints are considered. It is shown that the methods are…
The two-functional conjecture says that if a function f analytic and univalent in the unit disk maximizes Re{L} and Re{M} for two continuous linear functionals L and M, L is not equal to cM for any c>0, then f is a rotation of the Koebe…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper. The results are given in terms of $L^p$ integrability of the function and its…
We consider the conic linear program given by a closed convex cone in an Euclidean space and a matrix, where vector on the right-hand-side of the constraint system and the vector defining the objective function are subject to change. Using…
We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix…
In this work, several sharp bounds for the \v{C}eby\v{s}ev functional involving various type of functions are proved. In particular, for the \v{C}eby\v{s}ev functional of two absolutely continuous functions whose first derivatives are both…
We provide some necessary and sufficient conditions for a proper lower semicontinuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection…
We derive two upper bounds for the probability of deviation of a vector-valued Lipschitz function of a collection of random variables from its expected value. The resulting upper bounds can be tighter than bounds obtained by a direct…
We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies…