Related papers: Subdifferentials at infinity and applications in o…
Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to…
We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in…
Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…
Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the…
In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds…
The main contribution of this paper is that every convex function with non-empty relative algebraic interior of its domain is Lipschitz and subdifferentiable in some algebraic sense without any additional topological constraints. The…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
This work advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite…
This paper is devoted to the study of metric subregularity and strong subregularity of any positive order $q$ for set-valued mappings in finite and infinite dimensions. While these notions have been studied and applied earlier for $q=1$…
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…
This paper sheds new light on several interrelated topics of second-order variational analysis, both in finite and infinite-dimensional settings. We establish new relationships between second-order growth conditions on functions, the basic…
In this paper we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them called here rated extremal principles. These developments are in the…
The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various…
For various classes of Lipschitz functions we provide dimension free concentration inequalities for infinitely divisible random vectors with independent components and finite exponential moments.
Elementary facts and observations on the cone of supermodular set functions are recalled. The manuscript deals with such operations with set functions which preserve supermodularity and the emphasis is put on those such operations which…
In this paper, we present the concept of subdifferential for fuzzy n-cell number valued functions. Then we state some theorems related to subdifferentiability based on the new definition. Finally, we present some applications emphasized on…
We generalise the outer subdifferential constructon suggested by C\'anovas, Henrion, L\'opez and Parra for max type functions to pointwise minima of regular Lipschitz functions. We also answer an open question about the relation between the…
In many naturally occurring optimization problems one needs to ensure that the definition of the optimization problem lends itself to solutions that are tractable to compute. In cases where exact solutions cannot be computed tractably, it…
Lower and upper bounds for a given function are important in many mathematical and engineering contexts, where they often serve as a base for both analysis and application. In this short paper, we derive piecewise linear and quadratic…
Nearly convex sets play important roles in convex analysis, optimization and theory of monotone operators. We give a systematic study of nearly convex sets, and construct examples of subdifferentials of lower semicontinuous convex functions…