Concept Of Convexity Research Articles

Convex preferences: A new definition

We suggest a concept of convexity of preferences that does not rely on any algebraic structure. A decision maker has in mind a set of orderings interpreted as evaluation criteria. A preference relation is defined to be convex when it satisfies the following condition: If, for each criterion, there is an element that is both inferior to b by the criterion and superior to a by the preference relation, then b is preferred to a. This definition generalizes the standard Euclidean definition of convex preferences. It is shown that under general conditions, any strict convex preference relation is represented by a maxmin of utility representations of the criteria. Some economic examples are provided.

Convex optimization of interval valued functions on mixed domains

In this paper, we study a class of convex type interval-valued functions on the domain of the product of closed subsets of real numbers. By considering LW order relation on the class of closed intervals, we proposed some optimal solutions. LW convexity concepts and generalized Hukuhara differentiability (viz. delta and nabla) for interval-valued functions yield the necessary and sufficient conditions for interval programming problem. In addition, we compare our results with the results given in the literature. These results may open a new avenue for modeling and solve a different type of optimization problems that involve both discrete and continuous variables at the same time.

A Class of Conceptual Spaces Consisting of Boundaries of Infinite p-Ary Trees

A new construction of a certain conceptual space is presented. Elements of this conceptual space correspond to (and serve as code for) concept elements of reality, which potentially comprise an infinite number of qualities. This construction of a conceptual space solves a problem stated by Dietz and his co-authors in 2013 in the context of Voronoi diagrams. The fractal construction of the conceptual space is that this problem simply does not pose itself. The concept of convexity is discussed in this new conceptual space. Moreover, the meaning of convexity is discussed in full generality, for example when space is deprived of it, its substitutes for concept domains are considered.

Jensen\u2019s and Hermite\u2013Hadamard\u2019s Type Inequalities for Lower and Strongly Convex Functions on Normed Spaces

In this paper, we obtain some Jensen’s and Hermite–Hadamard’s type inequalities for lower, upper, and strongly convex functions defined on convex subsets in normed linear spaces. The case of inner product space is of interest since in these case the concepts of lower convexity and strong convexity coincides. Applications for univariate functions of real variable and the connections with earlier Hermite–Hadamard’s type inequalities are also provided.

Karush–Kuhn–Tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability

We extend the interval and fuzzy gH-differentiability to consider interval and fuzzy valued functions of several variables and to include directional gH-differentiability; the proposed setting is more general than the existing definitions in the literature and allows a unified view of total and direction gH-differentiability and for the computation of partial gH-derivatives, directional gH-derivative and level wise gH-differentiability in the fuzzy valued case. A concept of gH-differential is then deduced and its values are used to define an (abstract) local tangency property for a gH-differentiable function, similar to the well known tangency between a differentiable function and its tangent plane. The proposed new setting allows an analysis of conditions for local optimality (dominance with respect to interval and level-wise partial orders well known in the literature) in terms of directional gH-derivatives, including concepts of local convexity, and to formulate KKT-like conditions for non-dominated solutions in constrained optimization problems.

DIRECTIONAL CONVEXITY RECOGNITION OF THE UNION OF SEVERAL POLYHEDRA

The article is devoted to the algorithmic aspects of partial convexity, which is a generalization of the classical concept of convexity. Often, it is necessary to introduce the concept of partial convexity when studying many applied problems such as problems of VLSI design, image processing, database design, etc., since the requirement of traditional convexity is too restrictive for such problems. At the same time, the partial convexity preserves many useful properties of classical convexity, which makes it possible to find effective algorithms for solving these problems. A problem for recognizing directional convexity of the union of polyhedral sets in the n-dimensional linear space is investigated. We have established a necessary and sufficient attribute of the partial convexity for the union of polyhedral sets. Using this attribute of partial convexity, in the case of the finiteness of the set of directions of partial convexity, we have developed a polynomial recognition algorithm for the union of several polyhedral sets given by intersections of half-spaces, provided that the number of these sets is fixed. We note that earlier for the problem under consideration, a polynomial algorithm for solving it was known only for the case of the union of two polyhedral sets.

Optimality conditions for fuzzy constrained programming problems

This paper solves optimization problems where both the objective and constraints are given by fuzzy functions. In order to get it, we first prove that these problems are equivalent to optimization problems where the constraints functions are non-fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far. This new stationary point concept is based on the gH-differentiability and has many computational advantages that we describe. It is well-known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity. And we are in that case due to the fact that the fuzzy numbers (intervals) space is a nonlinear one. In this direction, the gH-derivative for fuzzy functions is a concept that is more general than Hukuhara and level-wise derivatives that are usually used in fuzzy optimization so far, in the sense that they can be applied to a wider number of fuzzy function classes than above concepts. With this new differentiability concept, we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far. Moreover, due to the fact that we do not have a linear space for fuzzy numbers, the convex concepts and generalized convex fuzzy function notion are very restrictive, also. This implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful.

A Newton-Like Method for Variable Order Vector Optimization Problems

A Newton approach is proposed for solving variable order smooth constrained vector optimization problems. The concept of strong convexity is presented, and its properties are analyzed. It is thus obtained that the Newton direction is well defined and that the algorithm converges. Moreover, the rate of convergence is obtained under ordering structures satisfying a mild hypothesis.

Monophonic convexity in weighted graphs

In this paper, the concept of monophonic convexity is extended to weighted graphs. Set theoretic operations of monophonic convex sets and some of their properties are discussed. The notions of weighted monophonic block, weighted monophonic boundary and interior points are also studied and their characterizations are obtained.

Asymptotic farthest points and extreme points

Let X be a normed space, G a nonempty bounded subset of X and fxng a bounded sequence in X. In this article, we introduce and discuss the concept of asymptotic farthest points of fxng in G, which is a new definition in abstract approximation theory. Then, by applying the topics of functional analysis, we investigate the relation between this new concept and the concepts of extreme points and convexity. In particular, one of the main purposes of this paper is to study conditions under which the existence (uniqueness) of asymptotic farthest point of fxng in G is equivalent to the existence (uniqueness) of asymptotic farthest point of fxng in ext(G) or co(G).

Bioinspired sprawling robotic leg and a path-planning procedure

This article shows the characteristics of a sprawling robotic leg inspired by the limb postures of certain reptilian animals known as sprawling-legged creatures. The main part of the robotic limb is based on the eight-bar Peaucellier–Lipkin linkage, and its main attribute is the ability to trace a true straight line, due to the rotational motion of the input link. However, when the eight-bar linkage is modified, it is capable of tracing true circular concave or convex arcs, when two of its constitutive links have distinct and precise lengths. This gives rise to the concepts of concavity and convexity, related to a robotic leg based on the Peaucellier–Lipkin mechanism, such as the one described herein. Our bioinspired robotic leg can trace concave or convex curves, as well as straight lines, making it a reptile-like robotic limb that is very similar to the natural one. We also introduce the concept of rotation center tuning, which refers to the ability of the leg to adapt its posture to the center of rotation of the entire walking machine, resulting in an easy and suitable gait process. The theoretical information is illustrated through the simulation of an example that provides a path-planning procedure, focusing on the rotation center tuning process and a walking gait. The example also includes the design of an elliptical path projected onto the cylindrical workspace and followed by the reptilian foot.

Generalization of the concept of convexity in a hypercomplex space

Extremal elements and a h-hull of sets in the n-dimensional hypercomplex space Hn are investigated. The class of H-quasiconvex sets including strongly hypercomplexly convex sets and closed relatively to intersections is introduced. Some results concerning multivalued functions in the complex space were generalized into the n-dimensional hypercomplex space: there was proved the hypercomplex analogue of the Fenchel-Moreau theorem and some properties of functions that are conjugate to functions f : Hn Ɵ a H.

Further on Set-Valued Equilibrium Problems and Applications to Browder Variational Inclusions

In this paper, we introduce some concepts of convexity and semicontinuity for real set-valued mappings similar to those of real single-valued mappings. Then, we obtain different results on the existence of solutions of set-valued equilibrium problems generalizing in a common way several old ones for both single-valued and set-valued equilibrium problems. Applications to Browder variational inclusions, with weakened conditions on the involved set-valued operator, are given.

Revisiting the Hahn–Banach theorem and nonlinear infinite programming

The aim of this paper is to state a sharp version of the König supremum theorem, an equivalent reformulation of the Hahn–Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush–Kuhn–Tucker and Fritz John types, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.

Perturbation of error bounds

Our aim in the current article is to extend the developments in Kruger, Ngai & Th\'era, SIAM J. Optim. 20(6), 3280-3296 (2010) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by proper lower semicontinuous under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the `radius of error bounds'. The definitions and characterizations are illustrated by examples.

Jensen's inequality type integral for fuzzy-interval-valued functions

This study presents fuzzy versions of Jensen inequalities type integral for convex and concave fuzzy-interval-valued functions. To this end, the concepts of fuzzy inclusion order relation, convexity, and concavity for fuzzy-interval-valued functions are used. Some examples showing the applicability of the theory developed in this study are presented. Since the fuzzy results are obtained through level sets of fuzzy-interval elements, the versions of these results in the interval context are presented here for the first time.

Discrete convex analysis: A tool for economics and game theory

This paper presents discrete convex analysis as a tool for use in economics and game theory. Discrete convex analysis is a new framework of discrete mathematics and optimization, developed during the last two decades. Recently, it has been recognized as a powerful tool for analyzing economic or game models with indivisibilities. The main feature of discrete convex analysis is the distinction of two convexity concepts, M-convexity and L-convexity, for functions in integer or binary variables, together with their conjugacy relationship. The crucial fact is that M-concavity in its variant is equivalent to the gross substitutes property in economics. Fundamental theorems in discrete convex analysis such as the M-L conjugacy theorems, discrete separation theorems and discrete fixed point theorems yield structural results in economics such as the existence of equilibria and the lattice structure of equilibrium price vectors. Algorithms in discrete convex analysis provide iterative auction algorithms for finding equilibria.

On an implicit convexity concept and some integral inequalities

We introduce a new concept of convexity that depends on a function $F:\mathbb{R}\times\mathbb{R}\times\mathbb {R}\times (0,1)\to\mathbb{R}$ satisfying certain axioms. The presented concept generalizes many kinds of convexity including e-convex functions, α-convex functions, and h-convex functions. Moreover, some integral inequalities are provided via our notion of convexity.

Methods for improving estimators of truncated circular parameters

In decision theoretic estimation of parameters in Euclidean space $\mathbb{R}^p$, the action space is chosen to be the convex closure of the estimand space. In this paper, the concept has been extended to the estimation of circular parameters of distributions having support as a circle, torus or cylinder. As directional distributions are of curved nature, existing methods for distributions with parameters taking values in $\mathbb{R}^p$ are not immediately applicable here. A circle is the simplest one-dimensional Riemannian manifold. We employ concepts of convexity, projection, etc., on manifolds to develop sufficient conditions for inadmissibility of estimators for circular parameters. Further invariance under a compact group of transformations is introduced in the estimation problem and a complete class theorem for equivariant estimators is derived. This extends the results of Moors [J. Amer. Statist. Assoc. 76 (1981) 910-915] on $\mathbb{R}^p$ to circles. The findings are of special interest to the case when a circular parameter is truncated. The results are implemented to a wide range of directional distributions to obtain improved estimators of circular parameters.

A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies

The series of papers published by Gil and Ortigosa (Gil and Ortigosa, 2016; Ortigosa and Gil, 2016, 0000) introduced a new convex multi-variable variational and computational framework for the numerical simulation of Electro Active Polymers (EAPs) in scenarios characterised by extreme deformations and/or extreme electric fields. Building upon this body of work, five key novelties are incorporated in this paper. First, a generalisation of the concept of multi-variable convexity to energy functionals additively decomposed into isochoric and volumetric components. This decomposition is typical of nearly and truly incompressible materials, group which represents the majority of the most relevant EAPs. Second, convexification or regularisation strategies are applied to a priori non-convex multi-variable isochoric functionals to yield physically meaningful convex multi-variable functionals. Third, based on the mixed variational principles introduced in Gil and Ortigosa (2016) in the context of compressible electro-elasticity, a novel extended Hu–Washizu mixed variational principle for nearly and truly incompressible scenarios is presented. From the computational standpoint, a static condensation procedure is applied in order to condense out the element-wise extra fields, the resulting formulation having a comparable cost to the more standard three-field displacement-potential-pressure mixed formulation. Fourth, the computational framework for the three-field mixed variational principle in nearly and truly incompressible scenarios is also presented. In this case, the novelty resides in the consideration of convex multi-variable energy functionals. Ultimately, this leads to the definition of new tangent operators for the Helmholtz’s energy functional in the specific context of incompressible electro-elasticity. Fifth, a Petrov–Galerkin stabilisation technique is applied on the three-field formulation for the circumvention of the Ladyz˘enskaja–Babus˘ka–Brezzi (LBB) condition, enabling the use of linear tetrahedral finite elements for the interpolation of the unknowns of the problem. Finally, a series of challenging numerical examples is presented in order to provide an exhaustive comparison of the different variational formulations presented in this paper.