Notion Of Convexity Research Articles

On polar convexity in finite-dimensional Euclidean spaces

Abstract Let $\hat {\mathbb {R}}^n$ be the one-point compactification of $\mathbb {R}^n$ obtained by adding a point at infinity. We say that a subset $A\subseteq \hat {\mathbb {R}}^n$ is $\mathbf {u}$ -convex if for every pair of points $\mathbf {z}_1, \mathbf {z}_2 \in A$ , the arc of the unique circle through $\mathbf {u}, \mathbf {z}_1$ and $\mathbf {z}_2$ , from $\mathbf {z}_1$ to $\mathbf {z}_2$ and not containing $\mathbf {u}$ , is contained in A. In this case, we call $\mathbf {u}$ a pole of A. When the pole $\mathbf {u}$ approaches infinity, $\mathbf {u}$ -convex sets become convex in the classical sense. The notion of polar convexity in the complex plane has been used to analyze the behavior of critical points of polynomials. In this paper, we extend the notion to finite-dimensional Euclidean spaces. The goal of this paper is to start building the theory of polar convexity and to show that the introduction of a pole creates a richer theory. For example, polar convexity enjoys a beautiful duality (see Theorem 4.3) that does not exist in classical convexity. We formulate polar analogues of several classical results of the alternatives, such as Gordan’s and Farkas’ lemmas; see Section 5. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles; see Theorem 6.7.

Sufficient convexity and best approximation

Working constructively throughout, we introduce the notion of sufficient convexity for functions and sets and study its implications on the existence of best approximations of points in sets and of sets mutually.

Correction: Revisited convexity notions for $$L^\infty$$ variational problems

Correction: Revisited convexity notions for $$L^\infty$$ variational problems

Polynomial convexity with degree bounds

We introduce different notions of polynomial convexity with bounds on degrees of polynomials in C n . We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree bounds for certain sets of points in C and for certain arcs in the unit circle.

Revisited convexity notions for $$L^\infty $$ variational problems

AbstractWe address a detailed study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals, as well as those arising by the $$L^p$$ L p -approximation, as $$p \rightarrow +\infty $$ p → + ∞ of such functionals. Our quest is motivated by the knowledge we have on the analogous integral functionals and aims at establishing a solid groundwork underlying further research in the $$L^\infty $$ L ∞ context.

Viscosity solutions of obstacle equations by inhomogeneous convex envelopes

We characterize the convex envelope of a continuous function as a viscosity solution of a second order differential equation of obstacle style. The notion of convexity that we investigate is an inhomogeneous generalization of the classical concept of convexity motivated by problems of singular control in finance.

Efficiency and the core in NTU games in partition function form

AbstractThe aim of this paper is to establish a necessary and sufficient condition for the non-emptiness of the core in NTU games in partition function form, given an externality scheme $$ f \in \textsf{Ex}(N) $$ f ∈ Ex ( N ) . We extend the notion of convexity to incorporate externality effects. By introducing a new concept of rationality, called collective rationality, we demonstrate the efficiency of the grand coalition $$ N $$ N . We also identify a sufficient condition for the efficiency of the grand coalition using the property of individual superadditivity.

Optimality and Unified Duality for E-differentiable Vector Optimization Problems over Cones involving Generalized E-convexity

This paper explores a new approach to solve a nondifferentiable vector optimization problem over cones by means of an operator E : Rn → Rn which renders differentiability to the considered problem. Some new generalized convexity notions are introduced and employed to obtain necessary and sufficient KKT-type optimality conditions. Further, a unified dual is associated with the considered problem encapsulating both Mond-Weir and Wolfe type duals in the same apparatus and duality results are proved.

Efficient targets and reference sets in selectively convex technologies

Conventional models of data envelopment analysis (DEA) typically assume that the underlying production technology is a convex set. It is known that such assumption may be clearly unsubstantiated in certain cases. Examples include studies in which some inputs or outputs are stated as proportions or percentages, or are represented by categorical measures. Excluding such “problematic” inputs and outputs from the assumption of convexity while assuming the latter for the remaining measures leads to the notion of selective convexity. Further examples of selectively convex technologies include technologies parameterized by an environmental factor and technologies in which only the input or output sets are convex. In this paper, we consider the identification of efficient targets and reference sets of decision making units in a selectively convex technology, which has not yet been explored in the literature. We show that, for such technologies, the conventional method based on the solution of the additive DEA model may not correctly identify the reference sets and needs an adjustment.

Generalized relative interiors and generalized convexity in infinite-dimensional spaces

This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We introduce, study, and utilize a novel notion of quasi-near convexity of sets that is an infinite-dimensional extension of the widely acknowledged notion of near convexity. Quasi-near convexity is associated with the quasi-relative interior of sets, which is investigated in the paper together with other generalized relative interior notions for sets, not necessarily convex. In this way, we obtain new results on generalized relative interiors for graphs of set-valued mappings in convexity and generalized convexity settings.

Spectral Total-variation Processing of Shapes—Theory and Applications

We present a comprehensive analysis of total variation (TV) on non-Euclidean domains and its eigenfunctions. We specifically address parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work sheds new light on the celebrated Beltrami and Anisotropic TV flows and explains experimental findings from recent years on shape spectral TV [Fumero et al. 2020 ] and adaptive anisotropic spectral TV [Biton and Gilboa 2022 ]. A new notion of convexity on surfaces is derived by characterizing structures that are stable throughout the TV flow, performed on surfaces. We establish and numerically demonstrate quantitative relationships between TV, area, eigenvalue, and eigenfunctions of the TV operator on surfaces. Moreover, we expand the shape spectral TV toolkit to include zero-homogeneous flows, leading to efficient and versatile shape processing methods. These methods are exemplified through applications in smoothing, enhancement, and exaggeration filters. We introduce a novel method that, for the first time, addresses the shape deformation task using TV. This deformation technique is characterized by the concentration of deformation along geometrical bottlenecks, shown to coincide with the discontinuities of eigenfunctions. Overall, our findings elucidate recent experimental observations in spectral TV, provide a diverse framework for shape filtering, and present the first TV-based approach to shape deformation.

A relaxation theorem for a first-order set differential inclusion in a metric space

Abstract The aim of this article is to present, in a specific metric space, a density theorem. Then, as an application, we give a relaxation theorem for a first-order set-differential inclusion, through an abstract convexity notion.

Umbel convexity and the geometry of trees

For every p∈(0,∞), a new metric invariant called umbel p-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local invariant which inspired it, captures the geometry of bounded degree trees. Umbel convexity is used to provide a “Poincaré-type” metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property (β). We explain how a relaxation of umbel p-convexity, called infrasup-umbel p-convexity, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogues of these invariants - fork p-convexity and infrasup-fork p-convexity - are introduced, and their relationship to Markov p-convexity and relaxations of the p-fork inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups, and in particular a parallelogram p-convexity inequality is proved for Heisenberg groups over p-uniformly convex Banach spaces. Finally, a new characterization of non-negative curvature is given.

Mean Field Approximations via Log-Concavity

Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $\mathbb {R}^{n}$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the log partition function $\log \int e^{f(x)}dx$ in terms of $\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$, for a semi-explicit probability measure $Q^{*}$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $H(\cdot \,|\,P)$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport.

On optimality for fuzzy optimization problems with granular differentiable fuzzy objective functions

In recent years, there has been a growing interest in optimization problems with uncertainty. Fuzzy optimization is one of the approaches for investigation of such real-world extremum problems that contain uncertain data and, therefore, that are not well defined. Nonetheless, there is not enough discussion on the optimality conditions for optimal solutions (with regard to distinct fuzzy numbers) in (nonconvex) granular differentiable fuzzy optimization problems and methods for solving such uncertain extremum problems. Although the convexity notion is a very important property of optimization models, there are many real-world extremum problems and processes with uncertainty that cannot be modeled as convex fuzzy optimization problems. In this work, therefore, a new notion of granular differentiable generalized convexity is defined. Namely, the definition of granular differentiable F-convexity is introduced for fuzzy optimization problems and some properties of this concept are analyzed. Further, both the Fritz John necessary optimality conditions and, under the Guignard constraint qualification, the Karush-Kuhn-Tucker necessary optimality conditions are established for an optimal solution (with respect to distinct fuzzy numbers) in a fuzzy extremum problem with granular differentiable fuzzy-valued objective function and both inequality and equality constraints. As main applications of the introduced concept of granular differentiable F-convexity, the sufficiency of the derived Karush-Kuhn-Tucker necessary optimality conditions are proved and a new approach based on the so-called modified F-objective function method is proposed for solving the investigated fuzzy extremum problem.

A variational theory for integral functionals involving finite-horizon fractional gradients

The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general varGamma -convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model.

Conceptual orthospaces—Convexity meets negation

Neural networks and in general subsymbolic learning approaches perform well on usual learning tasks, but they are black boxes lacking desired properties such as explainability or trustworthiness. Fully integrated symbolic/subsymbolic systems account for these properties, e.g., through the injection of symbolic information into a subsymbolic learning framework. Systems relying on embeddings can be considered as specific examples of such systems where facts, expressed in some logic, are embedded into a continuous space. Instances mentioned in the facts are mapped to points and the relations/concepts mentioned in the facts are mapped to a region in that space. This allows to exploit geometric regularities in order to tackle typical learning tasks such as link prediction. Embeddings provide the first step towards filling the gap between qualitative, Tarskian style semantics, which is used for deductive reasoning over the facts, and quantitative structures, which are used for representing objects, relations, and concepts for learning purposes. However, to enable meaningful reasoning, embeddings of relations and concepts are not allowed to be shaped arbitrarily. Especially, convex sets turned out to be appropriate due to their computational advantages and due to their foundation in cognition. Convexity can be defined with a ternary betweenness relation and concepts can be defined as betweenness-closed (= convex) sets. Though many interesting phenomena of cognitive reasoning can be explained in such a framework, it is at least not obvious how to use betweenness for other, more logico-formal aspects of reasoning that, e.g., require defining logical operators. In particular, for the logical operator of negation other mathematical structures such as the orthoframes of Goldblatt (1974) [13] have proven more useful. In this paper, we provide results on the connection between convexity and orthoframes. In particular, we investigate the construction of convex concepts equipped with an orthogonality relation. We give a universal construction of a betweenness relation over an orthoframe and show that based on Euclidean betweenness (thus using the classical notion of convexity) and under some natural restrictions, convex cones are the only structures capable of modeling such a space.

Distributed (ATC) Gradient Descent for High Dimension Sparse Regression

We study linear regression from data distributed over a network of agents (with no master node) by means of LASSO estimation, in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">high-dimension</i> , which allows the ambient dimension to grow faster than the sample size. While there is a vast literature of distributed algorithms applicable to the problem, statistical and computational guarantees of most of them remain unclear in high dimension. This paper provides a first statistical study of the Distributed Gradient Descent (DGD) in the Adapt-Then-Combine (ATC) form. Our theory shows that, under standard notions of restricted strong convexity and smoothness of the loss functions–which hold with high probability for standard data generation models–suitable conditions on the network connectivity and algorithm tuning, DGD-ATC converges globally at a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear</i> rate to an estimate that is within the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">centralized</i> statistical precision of the model. In the worst-case scenario, the total number of communications to statistical optimality grows logarithmically with the ambient dimension, which improves on the communication complexity of DGD in the Combine-Then-Adapt (CTA) form, scaling linearly with the dimension. This reveals that mixing gradient information among agents, as DGD-ATC does, is critical in high-dimensions to obtain favorable rate scalings.

Weighted average-convexity and Shapley values

We generalize the notion of convexity and average-convexity to the notion of weighted average-convexity. We show several results on the relation between weighted average-convexity and cooperative games. Our main result is that if a game is weighted average-convex, then the corresponding weighted Shapley value is in the core.

On some inequalities of Fejér’s type using the notion of harmonic convexity

In this study, weintroduce some new mappings in connection with Hermite-Hadamard and Fej?r type integral inequalities which have been proved using the harmonic convex functions. As a consequence, we discover certain new inequalities of the Fej?r type that provide refinements of the Hermite-Hadamard and Fej?r type integral inequalities that have already been obtained.