Convex Analysis

We examine the role of the convex structure in a metric space on which it is defined. First, we introduce the notion of extreme point and face of a convex set. Second, we present the idea of core in a convex metric space. Several properties are proved and examples to support are given.

In this work, we study the interplay of ordered relation and the structure of the convex metric space on which it is defined: First, we show when a convex metric space is isomorphic with a convex subset of some vector space. Second, we obtain conditions for a representation of a preference relation on a convex metric space.

It is well known that the construction of Voronoi diagrams is based on the notion of bisector of two given points. Already in normed linear spaces, bisectors have a complicated structure and can, for many classes of norms, only be described with the help of topological methods. Even more general, we present results on bisectors for convex distance functions (gauges). Let C, with the origin o from its interior, be the compact, convex set inducing a convex distance function (gauge) in the plane, and let B(-x, x) be the bisector of -x and x, i.e., the set of points z whose distance (measured with the convex distance function induced by C) to -x equals that to x. For example, we prove the following characterization of the Euclidean norm within the family of all convex distance functions: if the set L of points x in the boundary ∂C of C that create B(-x, x) as a straight line has non-empty interior with respect to ∂C, then C is an ellipse centered at the origin. For the subcase of normed planes we give an easier approach, extending the result also to higher dimensions.

Abstract. Min-independence has been proved to be a sufficient condition of a vector of fuzzy random variables to be a fuzzy random vector. The objective of this paper is to study further on the independence condition for fuzzy random vector based on continuous triangular ...

We develop a short and practical framework guaranteeing existence, uniqueness and stability of minimizers of functionals of the form E[u] = Ω W (∇u(x)) + V (u(x)) dx, where W and V satisfy explicit convexity and growth conditions expressed in terms of directly measurable parameters. Under these assumptions we establish: (i) coercivity of E, (ii) lower semicontinuity, (iii) existence and uniqueness of minimizers in W 1,p (Ω), and (iv) explicit a priori bounds. We provide short examples in nonlinear elasticity, composite materials and regularized imaging (Huber-TV) to illustrate how the proposed conditions are simpler to verify than classical quasiconvexity or polyconvexity requirements.

In this paper we study the connections between moduli of asymptotic convexity and smoothness of a Banach space, and the existence of high order differentiable bump functions or equivalent norms on the space. The existence of a high order uniformly differentiable bump function is related to an asymptotically uniformly smooth renorming of power type. On the other hand, the asymptotic uniform convexity of power type is related to the existence of high order rough norms. Finally, we also obtain some applications to the best order smoothness of Nakano sequence spaces.

Resumen: Approximation and rigidity properties in renorming constructions are characterized with some classes of simple maps. Those maps describe continuity properties up to a countable partition. The construction of such kind of maps can be done with ideas ...

In this paper we extend the results of recent studies on the existence of equilibrium in finite dimensional asset markets for both bounded and unbounded economies. We do not assume that the individual’s preferences are complete or transitive. Our existence theorems for asset markets allow for short selling. We shall also show that the equilibrium achieves a constrained core within the same framework.

We aim to present a theory for the derivation of relaxation operators in kinetic theory. The construction is based on an approximation of the inverse Boltzmann linearized operator, on relaxation equations on the moments of the distribution function and finally on a variational problem to be solved. The theory comprises a characterization of the set of moments of non negative integrable functions, a study of those linear application whose range lies in this set and a generalization of the functional to be minimized under moment constraints. In particular we clarify but also modify some steps in the proof of Junk's theorem on the characterization of moments of non negative functions . We also reestablish Csiszar's theorem by different means on a class of functionals leading to well-posed variational problems. The present theory encompasses the derivation of known models and that of new models.

For the first time in 2022, the authors introduced the notion of pseudo-Chebyshev wavelets in the context of one dimension. Continuing the study in advance sense, in this article, two dimensional pseudo Chebyshev wavelets are introduced. Two dimensional pseudo Chebyshev wavelet expansion of a function of two variable is defined and verified. This research paper introduces a novel algorithm based on the two dimensional pseudo Chebyshev wavelet method to address computation problems in approximation theory. The methods are illustrated by an example and compared with prominent Chebyshev wavelet methods to demonstrate the validity and applicability of the results. The error analysis and convergence analysis of a functions in the H ölder classes have been studied by this wavelets. More over the error of approximation of functions of Holder's class have been estimated by an orthogonal projection operators of its two dimensional pseudo Chebyshev wavelets. The results of this paper are the significant developments in wavelet analysis.

In this paper, we propose new extragradient algorithms for solving a split equilibrium and nonexpansive mapping SEPNM(C, Q, A, f , g, S, T) where C, Q are nonempty closed convex subsets in real Hilbert spaces H 1 , H 2 respectively, A : H 1 → H 2 is a bounded linear operator, f is a pseudomonotone bifunction on C and g is a monotone bifunction on Q, S, T are nonexpansive mappings on C and Q respectively. By using extragradient method combining with cutting techniques, we obtain algorithms for the problem. Under certain conditions on parameters, the iteration sequences generated by the algorithms are proved to be weakly and strongly convergent to a solution of this problem.

In this study, we establish some results for strong convergence of a sequence to a common fixed point of a subfamily of a nonexpansive and periodic evolution family of bounded linear operators acting on a closed and bounded subset J of a strictly convex Banach space X . In fact, we generalized the results from semigroups of the operator to an evolution family of operators.

The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact constraint and as such, is a convex problem with polynomial time complexity. Moreover, as a main pratical advantage of this relaxation over the standard Semi-Definite Programming approach, several efficient bundle methods are available for this problem allowing to address problems of very large dimension. The necessary tools from convex analysis are recalled and shown at work for handling the problem of exactness of this relaxation. Two applications are described. The first one is the problem of binary image reconstruction and the second is the problem of multiuser detection in CDMA systems.

We single out some second-order properties of convex functions that are well behaved with respect to the conjugacy operator. As an application, we prove that if a convex, lower semicontinuous function f: Iw" + iw v { + cc } has a second order 'Taylor expansion at the origin f(X)=f.4x.x+o(l~xJ2) as x+0, and the matrix A is symmetric and positive definite, then the conjugate functionf* has a second-order Taylor expansion at the origin, too, given by f*(Y)=fA-'Y~Y+o(l Yl? as y -0.

For a subgroup H of a topological abelian group G denote by group S(H) the set of all sequences of integers (u n ) such that u n h → 0 for every h ∈ H; H is called t-dense if S(H) = S(G). Motivated by a question of Raczkowski we explore the existence of small (with respect to size or measure) t-dense subgroups of topological abelian groups.

The problem of performing functional linear regression when the response variable is represented as a probability density function (PDF) is addressed. PDFs are interpreted as functional compositions, which are objects carrying primarily relative information. In this context, the unit integral constraint allows to single out one of the possible representations of a class of equivalent measures. On these bases, a function-on-scalar regression model with distributional response is proposed, by relying on the theory of Bayes Hilbert spaces. The geometry of Bayes spaces allows capturing all the key inherent features of distributional data (e.g., scale invariance, relative scale). A B-spline basis expansion combined with a functional version of the centred log-ratio transformation is utilized for actual computations. For this purpose, a new key result is proved to characterize B-spline representations in Bayes spaces. The potential of the methodological developments is shown on simulated data and a real case study, dealing with metabolomics data. A bootstrap-based study is performed for the uncertainty quantification of the obtained estimates.

This paper is intended as a first introduction into mathematical morphology, and does not require any preliminaryknowledge in this field. The paper discusses the basic morphological operators, such as dilations, erosions,openings, closings, rank operators, skeletons, and reconstruction operators. Furthermore, it introduces the completelattice framework for morphology. The paper concludes with an application of this framework to grey-scaleimages.1. Introduction

This paper presents a study of the morphological slope transform in the complete lattice framework. It discusses in detail the interrelationships between the slope transform at one hand and the (Young-Fenchel) conjugate and Legendre transform, two well-known concepts from convex analysis, at the other. The operators and transforms of importance here (hull operations, slope transform, support function, polar, gauge, etc.) are complete lattice operators with interesting properties also known from theoretical morphology. For example, the slope transform and its 'inverse' form an adjunction. It is shown that the slope transform for sets (binary signals) coincides with the notion of support Function, known from the theory of convex sets. Two applications are considered: the first application concerns an alternative approach to the distance transform. The second application deals with evolution equations for multiscale morphology using the theory of Hamilton-Jacobi equations. 0 1997 Elsevier Science B.V. In diesem Beitrag wird eine Untersuchung der morphologischen Steigungstransformation (morphological slope transform) im Rahmen der vollsttidigen lattice-Theorie prisentiert. Dabei werden die Zusarnmenhtige zwischen der Steigungstransformation einerseits und der (Young-Fenchel) konjugierten Transformation sowie der Legendre Transformation andererseits, die zwei wohlbekannte Verfahren aus der konvexen Analyse darstellen, diskutiert. Die Operatoren und Transformationen, die hier von Bedeutung sind (hull-Operationen, Steigungstransformation, support-Fur&ion, polar, gauge, etc.) stellen vollstiindige lattice-Operatoren mit interessanten Eigenschaften dar, die ebenfalls aus der theoretischen Morphologie bekannt sind. Die Steigungstransformation beispielsweise bildet zusammen mit ihrer Inversen ein adjungiertes Paar. Es wird gezeigt, dalj die Steigungstransformation fiir Mengen (bintie Signale) mit der Kenntnis der support-Funktion einhergeht, die aus der Theorie konvexer Mengen bekannt ist. Es werden zwei Anwendungen betrachtet: die erstc betrifft einen altemativen Ansatz zur Distanz-Transformation. Bei der zweiten geht es um Evolutionsgleichungen in der multiscale-Morphologie unter Ausnutzung der Theorie der Hamilton-Jacobi Gleichungen. 0 1997 Elsevier Science B.V. Cet article prksente une Ctude de la transformation de pente morphologique dans un cadre de structure en treillis. I1 discute en d&tail les inter-relations entre la transformation de pente d'un c8tC et les transformations conjuguCe (de Young-Fenchel) et de Legendre, deux concepts bien connus en analyse complexe, de l'autre. Les optrateurs et les transformations d'importance

We present some new methods for constructing convex functions. One of the methods is based on the composition of a convex function of several variables which is separately monotone with convex and concave functions. Using several well-known results on the composition of convex and quasi-convex functions we build new convex, quasi-convex, concave, and quasi-concave functions. The third section is dedicated to the study of convexity property of symmetric Archimedean functions. In the fourth section the asymmetric Archimedean function is considered. A classical example of such a function is the Bellman function. The fifth section is dedicated to the study of convexity/concavity of symmetric polynomials. In the sixth section a new proof of Chandler-Davis theorem is given. Starting from symmetric convex functions defined on finite dimensional spaces we build several convex functions of hermitian matrices. The seventh section is dedicated to a generalization of Muirhead's theorem and to some applications of it. The last section is dedicated to the construction of convex functions based on Taylor remainder series.

In this paper we present a review on the latest advances in logic-based solution methods for the global optimization of non-convex generalized disjunctive programs. Considering that the performance of these methods relies on the quality of the relaxations that can be generated, our focus is on the discussion of a general framework to find strong relaxations. We identify two main sources of non-convexities that any methodology to find relaxations should account for. Namely, the one arising from the non-convex functions and the one arising from the disjunctive set. We review the work that has been done on these two fronts with special emphasis on the latter. We then describe different logic-based optimization techniques that make use of the relaxation framework and its impact through a set of numerical examples typically encountered in Process Systems Engineering. Finally, we outline challenges and future lines of work in this area.

This paper considers the computational playability of backward induction solutions, $i.e$ . whether or not there is an algorithm to play them. We construct a two-person two-stage game with perfect information, in which both players have countably many feasible actions and their payoff functions are computable. We prove that the backward induction solutions of the game, which are proved to exist, are not computably playable because it is impossible to supply the players with algorithms regarding how they should play the solutions. Moreover, we show that if players' payoff functions are polynomial with rational coefficients, then the backward induction solutions are computably playable.

This paper considers the playability of noncooperative game solutions from the viewpoint of players' computational ability. We construct a two-person two-stage game with perfect information such that payoff functions are computable but no backward induction strategy is computable. Nevertheless we show the decidabil- ity of Nash equilibria of the game. These results mean that backward induction solutions cannot be played in practice because it is impossible to supply the players with effective instructions regarding how they should find the solutions, although Nash equilibria are playable.

The creation of goods and services requires changing the expended resources into the output goods and services. How efficiently we transform these input resources into goods and services depends on the productivity of the transformation process. However, it has been observed there is always a vagueness or imprecision associated with the values of inputs and outputs. Therefore, it becomes hard for a productivity measurement expert to specify the amount of resources and the outputs as exact scalar numbers. The present paper, applies fuzzy set theory to measure and compare productivity performance of transformation processes when numerical data cannot be specified in exact terms. The approach makes it possible to measure and compare productivity of organizational units (including non-government and non-profit entities) when the expert inputs can not be specified as exact scalar quantities. The model has been applied to compare productivity of different branches of a company.

We prove a tight asymptotic bound of Θ (δ log (n/δ)) on the worst case computational complexity of the convex hull of the union of two convex objects of sizes summing to n requiring δ orientation tests to certify the answer. For more convex objects, we prove a (non optimal) asymptotic bound of O (δ∑ ki= 1 log (ni/δ)) on the worst case computational complexity of the convex hull of the union of k convex objects of respective sizes (n1,..., nk) requiring δ orientation tests to certify the answer. Our algorithms are ...

Let R n denote the usual n-dimensional Euclidean space. A polyhedral convex function f : R n → R∪{+∞} can always be seen as the pointwise limit of a certain family {f t } t>0 of C ∞ convex functions. An explicit construction of this family {f t } t>0 can be found in a previous paper by the second author. The aim of the present work is to further explore this C ∞ -approximation scheme. In particular, one shows how the family {f t } t>0 yields first and second-order information on the behavior of f . Links to linear programming and Legendre-Fenchel duality theory are also discussed.

In this work, using the definitions of convex functions and h -convex functions, new Hermite-Hadamard type inequalities are presented using the framework of q -calculus. We prove inequalities for the q a -and q b -definite integrals of functions which have a convex or general convex q a -or q b -derivative. These inequalities have consequences for q -integrals and classical integrals, while extending some results previously known from the literature.

In this paper we will extend some properties of the convex real functions to thevalued functions in a Banach lattice: with adequate definitions, we will establishthat an order convex function is continuous on a convex C if and only if it iscontinuous at a point of C (Theorem 1.2). We will show that order convex functionson a compact satisfy Bauer’s maximal principle (Theorem 2.2). A fixed pointtheorem is given for the contracting orders functions (Theorem 2.3).

We first recall the basic elements of the non-associated (NA) Drucker-Prager plasticity model and then present the corresponding extended limit analysis theorems. Application of the latter to porous materials allows to establish a closed-form expression of the macroscopic criterion and of the nonassociated flow rule. The established results, for the porous material with a NA matrix, recover available models as particular cases. Finally, the new model is assessed and compares well with predictions by means of Finite Element computations carried out on a cell.

The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset $\Omega$ of a locally convex space $X$ and taking values in a locally convex space $Y$ ordered by a normal cone. One proves also equi-Lipschitz properties for pointwise bounded families of continuous convex mappings, provided the source space $X$ is barrelled. Some results on Lipschitz properties of continuous convex functions defined on metrizable topological vector spaces are included as well. The paper has a methodological character - its aim is to show that some geometric properties (monotonicity of the slope, the normality of the seminorms) allow to extend the proofs from the scalar case to the vector one. In this way the proofs become more transparent and natural.

We show that the value function of an optimal stopping game driven by a one-dimensional diffusion can be characterised using the extension of the Legendre transform introduced in . It is shown that under certain integrability conditions, a Nash equilibrium of the optimal stopping game can be derived from this extension of the Legendre transform. This result is an analytical complement to the results in where the 'duality' between a concave biconjugate which is modified to remain below an upper barrier and a convex biconjugate which is modified to remain above a lower barrier is proven by appealing to the probabilistic result in . The main contribution of this paper is to show that, for optimal stopping games driven by a one-dimensional diffusion, the semiharmonic characterisation of the value function may be proven using only results from convex analysis.

Illumination changes cause serious problems in many computer vision applications. We present a new method for addressing robust depth estimation from a stereo pair under varying illumination conditions. First, a spatially varying multiplicative model is developed to account for brightness changes induced between left and right views. The depth estimation problem, based on this model, is then formulated as a constrained optimization problem in which an appropriate convex objective function is minimized under various convex constraints modelling prior knowledge and observed information. The resulting multiconstrained optimization problem is finally solved via a parallel block iterative algorithm which offers great flexibility in the incorporation of several constraints. Experimental results on both synthetic and real stereo pairs demonstrate the good performance of our method to efficiently recover depth and illumination variation fields, simultaneously.

Qualitative and quantitative aspects for variational inequalities governed by merely continuous and strongly pseudomonotone operators are investigated in this paper. First, we establish a global error bound for the solution set of the given problem with the residual function being the normal map. Second, we will prove that the iterative sequences generated by gradient projection method (GPM) with stepsizes forming a non-summable diminishing sequence of positive real numbers converge to the unique solution of the problem with bounded constraint set. Two counter-examples are given to show the necessity of the boundedness assumption and the variation of stepsizes. A modification of GPM is proposed for unbounded case. Finally, we analyze the convergence rate of the iterative sequences generated by this method.

In this paper, the concept of strongly h-convex stochastic processes is introduced. New inequality related to Hermite-Hadamard type for strongly h-convex stochastic processes is obtained. Some properties of convex stochastic processes are presented. The results given in this work are generalizations of the known results.

The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which are described by inequality constraints which are locally Lipschitz and not necessarily convex and need not be smooth. We show that if the Slater's constraint qualification and a simple non-degeneracy condition is satisfied then the Karush-Kuhn-Tucker type optimality condition is both necessary and sufficient.

We define the concept of completion for locally convex cones. We show that how a locally convex cone with (SP) can be embedded as an upper dense subcone in an upper complete locally convex cone with (SP). We prove that every upper complete locally convex cone with (SP) is also symmetric complete.

An n-convex function is one whose nth order divided differences are nonnegative. Thus a l-convex function is nondecreasing and a 2-convex function is convex in the classical sense. A function f is n-concave if -f is n-convex. We consider best uniform approximation by n-convex and n-concave functions and, by considering alternation properties of the error function, we prove a variety of results, including characterizations of functions with best n-convex or n-concave approximations in fl,-i, a sufficient condition for best n-convex approximation, and a uniqueness result. Several examples are given.

We present some new methods for constructing convex functions. One of the methods is based on the composition of a convex function of several variables which is separately monotone with convex and concave functions. Using several well-known results on the composition of convex and quasi-convex functions we build new convex, quasi-convex, concave, and quasi-concave functions. The third section is dedicated to the study of convexity property of symmetric Archimedean functions. In the fourth section the asymmetric Archimedean function is considered. A classical example of such a function is the Bellman function. The fifth section is dedicated to the study of convexity/concavity of symmetric polynomials. In the sixth section a new proof of Chandler-Davis theorem is given. Starting from symmetric convex functions defined on finite dimensional spaces we build several convex functions of hermitian matrices. The seventh section is dedicated to a generalization of Muirhead's theorem and to some applications of it. The last section is dedicated to the construction of convex functions based on Taylor remainder series.

The regular N -gon provides the minimal Cheeger constant in the class of all N -gons with fixed volume. This result is due to a work of Bucur and Fragalà in 2014. In this note, we address the stability of their result in terms of the L 1 distance between sets. Furthermore, we provide a stability inequality in terms of the Hausdorff distance between the boundaries of sets in the class of polygons having uniformly bounded diameter. Finally, we show that our results are sharp, both in the exponent of decay and in the notion of distance between sets.

We reformulate the Cheeger N partition problem as a minimization among a suitable class of BV functions. This allows us to obtain a new existence proof for the Cheeger-N-problem. Moreover, we derive some connections between the Cheeger-2- problem and the second eigenvalue of the 1-Laplace operator.

Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined on the space of functions with Bounded Deformation. Mild additional continuity assumptions are required on the functionals.

We have established this paper on m-convex functions, which can be expressed as a general form of the convex function concept. First of all, some inequalities of Hadamard type are proved with fairly simple conditions. Next, an integral identity containing Atangana-Baleanu fractional integral operators is obtained to prove new inequalities for differentiable mconvex functions. Using this identity, various properties of m-convex functions and classical inequalities, some new integral inequalities have been proved.

In this note we study four different cone concepts used in the recent literature. In the case of no half lines in indifference surfaces (NHL) we show that the arbitrage cone, the recession cone of the preferred set, coincides with the Page-Wooders increasing cone and with the closure of Chichilnisky's original global cone (cf., Chichilnisky 1995). The fact that a closure is required for this equivalence underlies the difficulties with Chichilnisky's purported existence of equilibrium results using the global cone. From the equivalence of the increasing cone and the arbitrage cone in the NHL case, it follws that the existence of equilibrium result claimed in Chichilnisky (1997) is a special case of a more general result due to . Under some special assumptions, in the half lines (HL) case, where the normalized set of gradients to each indifference surface is a closed set, we demonstrate that the global cone is determined by the union of the directions of the half lines in indifference surfaces. Under the same special assumptions for the HL case, the increasing cone, the global cone and the interior of the arbitrage cone all coincide. We identify a gap in Chichilnisky's purported proof of existence of equilibrium in the HL case and discuss additional conditions that will enable a proof of her conjecture for this case.

A typical compact starshaped set in Ed is "small" from the topological as well as from the measure theoretic viewpoint. We formulate this more explicitly in the paper by using the notions of porosity and Hausdorff dimension. Moreover, we see that the directions of the line segments in a typical compact starshaped set are many, but not too many.