21,324 research outputs found
Generalizations of nondifferentiable convex functions and some characterizations
In this paper we generalize the convex functions, defining the concept of
preconvex function and we study some characterizations by intervals, some characterizations by polytopes, some characterizations by level sets, some properties of the extreme points and some relations whith the convex functions. Also, we define the R-quasiconvex functions as a generalization of the quasiconvex functions, and we study some characterizations by level sets and by separation sets, and some relations with the quasiconvex functions
Convexity of sets and quadratic functions on the hyperbolic space
In this paper some concepts of convex analysis on hyperbolic space are
studied. We first study properties of the intrinsic distance, for instance, we
present the spectral decomposition of its Hessian. Next, we study the concept
of convex sets and the intrinsic projection onto these sets. We also study the
concept of convex functions and present first and second order
characterizations of these functions, as well as some optimization concepts
related to them. An extensive study of the hyperbolically convex quadratic
functions is also presented
Recent Progress on Integrally Convex Functions
Integrally convex functions constitute a fundamental function class in
discrete convex analysis, including M-convex functions, L-convex functions, and
many others. This paper aims at a rather comprehensive survey of recent results
on integrally convex functions with some new technical results. Topics covered
in this paper include characterizations of integral convex sets and functions,
operations on integral convex sets and functions, optimality criteria for
minimization with a proximity-scaling algorithm, integral biconjugacy, and the
discrete Fenchel duality. While the theory of M-convex and L-convex functions
has been built upon fundamental results on matroids and submodular functions,
developing the theory of integrally convex functions requires more general and
basic tools such as the Fourier-Motzkin elimination.Comment: 50 page
Notes on Concavity, Convexity, Quasiconcavity and Quasiconvexity
This is just a quick and condensed note on the basic definitions and characterizations of concave, convex, quasiconcave and (to some extent) quasiconvex functions, with some examples
Characterizations of Nonemptiness and Compactness of the Set of Weakly Efficient Solutions for Convex Vector Optimization and Applications
AbstractIn this paper, we give characterizations for the nonemptiness and compactness of the set of weakly efficient solutions of an unconstrained/constrained convex vector optimization problem with extended vector-valued functions in terms of the 0-coercivity of some scalar functions. Finally, we apply these results to discuss solution characterizations of a constrained convex vector optimization problem in terms of solutions of a sequence of unconstrained vector optimization problems which are constructed with a general nonlinear Lagrangian
Evenly convex sets, and evenly quasiconvex functions, revisited
Since its appearance, even convexity has become a remarkable notion in convex analysis. In the fifties, W. Fenchel introduced the evenly convex sets as those sets solving linear systems containing strict inequalities. Later on, in the eighties, evenly quasiconvex functions were introduced as those whose sublevel sets are evenly convex. The significance of even convexity relies on the different areas where it enjoys applications, ranging from convex optimization to microeconomics. In this paper, we review some of the main properties of evenly convex sets and evenly quasiconvex functions, provide further characterizations of evenly convex sets, and present some new results for evenly quasiconvex functions.This research has been partially supported by MINECO of Spain and ERDF of EU, Grants PGC2018-097960-B-C22 and ECO2016-77200-P
Algebraic Properties of Arbitrage: An Application to Additivity of Discount Functions
Background: This paper aims to characterize the absence of arbitrage in the context of the Arbitrage Theory proposed by Kreps (1981) and Clark (2000) which involves a certain number of well-known financial markets. More specifically, the framework of this model is a linear (topological) space X in which a (convex) cone C defines a vector ordering. There exist markets for only some of the contingent claims of X which assign a price pi to the marketed claim mi . The main purpose of this paper is to provide some novel algebraic characterizations of the no arbitrage condition and specifically to derive the decomposability of discount functions with this approach. Methods: Traditionally, this topic has been focused from a topological or probabilistic point of view. However, in this manuscript the treatment of this topic has been by using purely algebraic tools. Results: We have characterized the absence of arbitrage by only using algebraic concepts, properties and structures. Thus, we have divided these characterizations into those concerning the preference relation and those involving the cone. Conclusion: This paper has provided some novel algebraic properties of the absence of arbitrage by assuming the most general setting. The additivity of discount functions has been derived as a particular case of the general theory
a classroom note on twice continuously differentiable strictly convex and strongly quasiconvex functions
We provide some remarks and clarifications for twice continuously differentiable strictly convex and strongly quasiconvex functions. Characterizations of these classes and their relationships with other classes of generalized convex functions are also examined
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