New Order Relations in Set Optimization

In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set P(Y)\mathcal{P}(Y) of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order \preccurlyeq\preccurlyeq: minimal elements, semicompactness, completeness, domination property of a subset of Q\mathcal{Q}, and semicontinuity of a set-valued map with values in Q\mathcal{Q} in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) from which one can easily derive similar results for the case, when F takes values on P(Y)\mathcal{P}(Y) equipped with various order relations.     

Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] ^G of a three dimensional representation W of G where G ? SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] ^G . The case where W is any two dimensional representation of G is also handled.     

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.     

Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces

The aim of this paper is to establish the semilocal convergence of a multipoint third order Newton-like method for solving F(x)=0 in Banach spaces by using recurrence relations. The convergence of this method is studied under the assumption that the second Fréchet derivative of F satisfies Hölder continuity condition. This continuity condition is milder than the usual Lipschitz continuity condition. A new family of recurrence relations are defined based on the two new constants which depend on the operator F. These recurrence relations give a priori error bounds for the method. Two numerical examples are worked out to demonstrate the applicability of the method in cases where the Lipschitz continuity condition over second derivative of F fails but Hölder continuity condition holds.     

Higher order optimality conditions with an arbitrary non-differentiable function

In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Fréchet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order n.     

Folded fans and string functions

Folded fans Fψ describe recursion properties of weights of integrable highest weight modules L ^u . Being considered simultaneously for the set of string functions s_s^u \sigma_s^u belonging to a fundamental Weyl chamber and corresponding to the same congruence class, the system of recursion relations gives rise to an equation which connects the string functions and the power series depending on multiplicities of weights of a folded fan Fψ. We apply these equations to study properties of string functions s_s^u \sigma_s^u associated with integrable modules for affine Lie algebras. New important relations for string functions are thus obtained. The set of folded fans provides a compact and effective tool to study them. Bibliography: 7 titles.     

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}We attempt a broad exploration of properties and connections between the symmetry function of a convex set S and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point , let sym(x,S) denote the symmetry value of x in S: , which essentially measures how symmetric S is about the point x, and define x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the L?wner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory. Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.     

Some relationships among gw-subdifferential,directional derivative and radial epiderivative for nonconvex vector functions

Abstract

In this work, we give some characterizations of gw-subdifferentiability of a vector-valued function by using its directional derivative and radial epiderivative. Moreover, under some assumptions, we proved that the directional derivative and radial epiderivative of a vector-valued function are the elements of the supremum set of gw-subgradients of it. Finally, without any convexity assumption, we proved that the epigraph of contingent derivative of a set valued map is included in the epigraph of contingent epiderivative of this set-valued map.     

On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and C ^1,1 functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.     

An Operation Connected to a Young-Type Inequality

Given two φ-functions F and G we consider the largest φ-function H = F ⊕ G such that the Young-type inequality H(xy) ? F(x) + G(y) holds for all x, y > 0. We prove an equivalence theorem for F ⊕ G with the best constants and, for the special case when F and G are log-convex and satisfy a certain growth condition, a representation formula for F G. Moreover, further properties and examples are presented and the relations to similar results are discussed.     

Constructing a sequence of discrete Hessian matrices of an SC 1 function uniformly convergent to the generalized Hessian matrix

We construct a uniform approximation for generalized Hessian matrix of an SC ¹ function. Using the discrete gradient and the extended second order derivative, we define the discrete Hessian matrix. We construct a sequence of sets, where each set is composed of discrete Hessian matrices. We first show some new properties of SC ¹ functions. Then, we prove that for SC ¹ functions the sequence of the set of discrete Hessian matrices is uniformly convergent to the generalized Hessian matrix.      

Representations of algebraic groups over a 2-dimensional local field

We introduce a categorical framework for the study of representations of G(F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G(F), which is an object of a suitable category of representations of G(F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G(F), when we consider the action by right translations.     

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category F≀FIn

We give a new proof for the Littlewood-Richardson rule for the wreath product F?S_n where F is a finite group. Our proof does not use symmetric functions but use more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F?S_n to F?S_n+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F?FI_n where FI_n is the category of all injective functions between subsets of an n-element set.     

Perturbations and Metric Regularity

A point x is an approximate solution of a generalized equation b∈F(x) if the distance from the point b to the set F(x) is small. ‘Metric regularity’ of the set-valued mapping F means that, locally, a constant multiple of this distance bounds the distance from x to an exact solution. The smallest such constant is the ‘modulus of regularity’, and is a measure of the sensitivity or conditioning of the generalized equation. We survey recent approaches to a fundamental characterization of the modulus as the reciprocal of the distance from F to the nearest irregular mapping. We furthermore discuss the sensitivity of the regularity modulus itself, and prove a version of the fundamental characterization for mappings on Riemannian manifolds. Mathematics Subject Classifications 2000 Primary: 49J53; secondary: 90C31.     

The disjunctive domination number of a graph

Abstract

For a positive integer b, we define a set S of vertices in a graph G as a b-disjunctive dominating set if every vertex not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it. The b-disjunctive domination number is the minimum cardinality of such a set. This concept is motivated by the concepts of distance domination and exponential domination. In this paper, we start with some simple results, then establish bounds on the parameter especially for regular graphs and claw-free graphs. We also show that determining the parameter is NP-complete, and provide a linear-time algorithm for trees.     

Sturmian morphisms,the braid group <Emphasis Type="Italic">B</Emphasis><Subscript>4</Subscript>, Christoffel words and bases of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F ₂) of automorphisms of the rank two free group F ₂ and show that it can be realized as a monoid in the group B ₄ of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F ₂ lifting any given basis of the free abelian group Z ². We further give an algorithm allowing to decide whether two elements of F ₂ form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15     

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

   Abstract. Let F be a family of disjoint unit balls in R ³ . We prove that there is a Helly-number n ₀ ≤ 46 , such that if every n ₀ members of F ( | F | ≥ n ₀ ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.     

Metric Distance Function and Profit: Some Duality Results

In this paper, we intend to establish relations between the way efficiency is measured in the literature on efficiency analysis and the notion of distance in topology. To this effect, we are interested particularly in the Hölder norm concept, providing a duality result based upon the profit function. Along this line, we prove that the Luenberger shortage function and the directional distance function of Chambers, Chung, and Färe appear as special cases of some l _p distance (also called Hölder distance), under the assumption that the production set is convex. Under a weaker assumption (convexity of the input correspondence), we derive a duality result based on the cost function, providing several examples in which the functional form of the production set is specified.     

A set scalarization function based on the oriented distance and relations with other set scalarizations

ABSTRACT

The aim of this paper is, in the setting of normed spaces with a cone K non necessarily solid, to study new relations among set scalarization functions that are extensions of the oriented distance of Hiriart-Urruty. Moreover, we deal with a set scalarization function of sup-inf type, we investigate its relation to the cone-properness and cone-boundedness and it is related to other set scalarizations existing in the literature. In particular, with the norm induced by the Minkowski's functional, we obtain relations with a set scalarization which is an extension of the so called Gerstewitz's scalarization function.     

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

Abstract. Let F be a family of disjoint unit balls in R ³ . We prove that there is a Helly-number n ₀ ≤ 46 , such that if every n ₀ members of F ( | F | ≥ n ₀ ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.