Results 1 to 10 of about 531,799 (95)
An extension of the proximal point algorithm beyond convexity [PDF]
Journal of Global Optimization, 2021, 2021 We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains ...
arxiv +1 more source
Inclusion and Intersection Relations Between Fundamental Classes of Discrete Convex Functions [PDF]
arXiv, 2021 In discrete convex analysis, various convexity concepts are considered for discrete functions such as separable convexity, L-convexity, M-convexity, integral convexity, and multimodularity. These concepts of discrete convex functions are not mutually independent.
arxiv
Shapley-Folkman-type Theorem for Integrally Convex Sets [PDF]
arXiv, 2023 The Shapley-Folkman theorem is a statement about the Minkowski sum of (non-convex) sets, expressing the closeness of the Minkowski sum to convexity in a quantitative manner. This paper establishes similar theorems for integrally convex sets, L-natural-convex sets, and M-natural-convex sets, which are major classes of discrete convex sets in discrete ...
arxiv
Discrete Fenchel Duality for a Pair of Integrally Convex and Separable Convex Functions [PDF]
arXiv, 2021 Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min-max theorem for a pair of integer-valued M-natural-convex functions generalizes the min-max formulas for polymatroid intersection and valuated matroid intersection.
arxiv
Polynomially convex sets whose union has nontrivial hull [PDF]
arXiv, 2021 Several results concerning pairs of polynomially convex sets whose union is not even rationally convex are given. It is shown that there is no restriction on how two spaces can be embedded in some $\C^N$ so as to be polynomially convex but have nonrationally convex union.
arxiv
A new treatment of convex functions [PDF]
arXiv, 2020 Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $\log-$convexity.
arxiv
A geometric approach to second-order differentiability of convex functions [PDF]
arXiv, 2023 We show a new, elementary and geometric proof of the classical Alexandrov theorem about the second order differentiability of convex functions. We also show new proofs of recent results about Lusin approximation of convex functions and convex bodies by $C^{1,1}$ convex functions and convex bodies.
arxiv
Sharp boundary regularity for some degenerate-singular Monge-Ampère Equations on k-convex domain [PDF]
arXiv, 2023 We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Amp\`ere type equations and prove the sharp boundary estimates for convex viscosity ...
arxiv
Decomposition of an Integrally Convex Set into a Minkowski Sum of Bounded and Conic Integrally Convex Sets [PDF]
arXiv, 2023 Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as integrally convex sets, L-natural-convex sets, and M-natural-convex sets.
arxiv
Preliminaries on CAT (0) Spaces and Fixed Points of a Class of Iterative Schemes [PDF]
arXiv, 2016 This paper gives some relating results for various concepts of convexity in metric spaces such as midpoint convexity, convex structure, uniform convexity and near-uniform convexity, Busemann curvature and its relation to convexity. Some properties of uniform convexity and near uniform convexity of geodesic metric spaces are related to the mapping built
arxiv