Efficient and weakly efficient solutions for the vector topical optimization

In this paper, we consider some scalarization functions, which consist of the generalized min-type function, the so-called plus-Minkowski function and their convex combinations. We investigate the abstract convexity properties of these scalarization functions and use them to identify the maximal points of a set in an ordered vector space. Then, we establish some versions of Farkas type results for the infinite inequality system involving vector topical functions. As applications, we obtain the necessary and sufficient conditions of efficient solutions and weakly efficient solutions for a vector topical optimization problem, respectively.     

Optimization of the difference of topical functions

In this paper, we first obtain some properties of topical (increasing and plus-homogeneous) functions in the framework of abstract convexity. Next, we use the Toland–Singer formula to characterize the dual problem for the difference of two topical functions. Finally, we present necessary and sufficient conditions for the global minimum of the difference of two strictly topical functions.     

A Study of Downward Sets with p-Functions

In this paper, we study downward sets and increasing functions in a topological vector space and their similarities to the convex sets and convex functions. It will be shown that a very special increasing function, namely, the p-function, can give a geometric interpretation for separating downward sets from outside points. Also, this function can be used to approximate topical functions in the framework of abstract convexity.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

Generalized vector quasi-equilibrium problems with applications

In this paper, we consider the generalized vector quasi-equilibrium problem with or without involving Φ-condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems. As applications of our results, we derive some existence results for a solution to the vector quasi-optimization problem for nondifferentiable functions and vector quasi-saddle point problem.     

Vector variational inequality with pseudoconvexity on Hadamard manifolds

In this paper, we give some properties for nondifferentiable pseudoconvex functions on Hadamard manifolds, and discuss the connections between pseudoconvex functions and pseudomonotone vector fields. Moreover, we study Minty and Stampacchia vector variational inequalities, which are formulated in terms of Clarke subdifferential for nonsmooth functions. Some relations between the vector variational inequalities and nonsmooth vector optimization problems are established under pseudoconvexity or pseudomonotonicity. The results presented in this paper extend some corresponding known results given in the literatures.     

The System of Vector Quasi-Equilibrium Problems with Applications

In this paper, we consider the system of vector quasi-equilibrium problems with or without involving -condensing maps and prove the existence of its solution. Consequently, we get existence results for a solution to the system of vector quasi-variational-like inequalities. We also prove the equivalence between the system of vector quasi-variational-like inequalities and the Debreu type equilibrium problem for vector-valued functions. As an application, we derive some existence results for a solution to the Debreu type equilibrium problem for vector-valued functions.     

On the Weak Semi-continuity of Vector Functions and Minimum Problems

Lower semi-continuity from above or upper semi-continuity from below has been used by many authors in recent papers. In this paper, we first study the weak semi-continuity for vector functions having particular form as that of Browder in ordered normed vector spaces; we obtain several new results on the lower semi-continuity from above or upper semi-continuity from below for these vector functions. Our results generalize some well-known results of Browder in scalar case. Secondly, we study the minimum or maximum problems for vector functions satisfying lower semi-continuous from above or upper semi-continuous from below conditions; several new results on the existence of minimal points or maximal points are obtained. We also use these results to study vector equilibrium problems and von Neumann’s minimax principle in ordered normed vector spaces.     

Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems

In this paper, we consider a class of split mixed vector quasivariational inequality problems in real Hilbert spaces and establish new gap functions by using the method of the nonlinear scalarization function. Further, we obtain some error bounds for the underlying split mixed vector quasivariational inequality problems in terms of regularized gap functions. Finally, we give some examples to illustrate our results. The results obtained in this paper are new.     

Quasispaces induced by vector fields measurable in ℝ3

We study some metric functions that are induced by a class of basis vector fields in ?³ with measurable coordinates. These functions are proved to be quasimetrics in the domain of definition of the vector fields. Under some natural constraints, the Rashevsky-Chow Theorem and the Ball-Box Theorem are established for the classes of vector fields we consider.     

On Minty variational principle for nonsmooth vector optimization problems with generalized approximate convexity

In this paper, we consider a nonsmooth vector optimization problem involving locally Lipschitz generalized approximate convex functions and find some relations between approximate convexity and generalized approximate convexity. We establish relationships between vector variational inequalities and nonsmooth vector optimization problem using the generalized approximate convexity as a tool.     

On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.     

Extended Farkas lemma and strong duality for composite optimization problems with DC functions

In this paper, we consider the optimization problem in locally convex Hausdorff topological vector spaces with objectives given as the difference of two composite functions and constraints described by an arbitrary (possibly infinite) number of convex inequalities. Using the epigraph technique, we introduce some new constraint qualifications, which completely characterize the Farkas lemma, the dualities between the primal problem and its dual problem. Applications to the conical programming with DC composite function are also given.     

An algebra of differential operators and generating functions on the set of univalent functions

With a method close to that of Kirillov [4], we define sequences of vector fields on the set of univalent functions and we construct systems of partial differential equations which have the sequence of the Faber polynomials (F_n) as a solution. Through the Faber polynomials and Grunsky coefficients, we obtain the generating functions for some of the sequences of vector fields.     

Order-Preserving Transformations and Applications

In this paper, we study the effects of a linear transformation on the partial order relations that are generated by a closed and convex cone in a finite-dimensional space. Sufficient conditions are provided for a transformation preserving a given order. They are applied to derive the relationship between the efficient set of a set and its image under a linear transformation, to characterize generalized convex vector functions by using order-preserving transformations, to establish some calculus rules for the subdifferential of a convex vector function, and develop an optimality condition for a convex vector problem.     

Betti numbers in multidimensional persistent homology are stable functions

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of scalar‐valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, that is, the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector‐valued filtering functions, we can consider the multidimensional analog of persistent Betti numbers. Varying the lower level sets, we obtain that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non‐negative integers. In this paper, we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector‐valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector‐valued filtering functions and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalar‐valued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers, we obtain a lower bound for the natural pseudo‐distance. Copyright © 2013 John Wiley & Sons, Ltd.     

On Hölder continuity of solution maps to parametric vector primal and dual equilibrium problems

In this paper, we first propose some kinds of the strong convexity (concavity) for vector functions. Then we apply these assumptions to establish sufficient conditions for the Hölder continuity of solution maps of the vector primal and dual equilibrium problems in metric linear spaces. As applications, we derive the Hölder continuity of solution maps of vector optimization problems and vector variational inequalities. Our results improve and generalize some recent existing ones in the literature.     

Abstract convexity of topical functions

In this paper, we examine properties of topical (increasing and plus-homogeneous) functions defined on a normed linear space ${X}$ . We also study many results of abstract convexity such as support set, polarity and subdifferential set of these functions. Finally, we give a characterization for topical functions with respect to closed downward sets.     

On Generalized Vector Equilibrium Problems

In this paper we prove the existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces. As applications, we present some relevant particular cases: a generalized vector variational-like inequality in Hausdorff topological vector spaces, and equilibrium problem in the case of pseudomonotone real functions, and a generalized weak Pareto optima problem.