Closed Convex Cone Research Articles

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.

Group majorization, eaton triples and numerical range*

Let G be a closed subgroup of the orthogonal group on a finite dimensional real inner product space V. The triple(V,G,F) is an Eaton triple if F≌V is a nonempty closed convex cone such that is nonempty for each x∈V. for all x,y∈F The Eaton triple (W,H,F)is called a reduced triple of the Eaton triple (V,G,F) if . In this event, by using a recent result of Niezgoda, a generaliszation of a result of Thompson and Poon on the convex hull of the G-orbit is given. Some matrix inequalities of Ky Fan on the sum of matrices are also extended. A generalized numerical range is introduced and we extend some results of Au-Yeung and Tsing on the equivalent statements of the convexity of the generalized numerical range associated with some Lie group. A question raised in [21]is answered by using a polygonal path result of Eaton and Perlman. We also discuss a possible extension of Schur-Horn-Kostant convexity result. Some questions are asked. The studies are motivated by some existing Lie theoretical results.

Understanding the Geometry of Infeasible Perturbations of a Conic Linear System

We discuss some properties of the distance to infeasibility of a conic linear system Ax = b, \; x \in C, $ where C is a closed convex cone. Some interesting connections between the distance to infeasibility and the solution of certain optimization problems are established. Such connections provide insight into the estimation of the distance to infeasibility and the explicit computation of infeasible perturbations of a given system. We also investigate the properties of the distance to infeasibility assuming that the perturbations are restricted to have a particular structure. Finally, we extend most of our results to more general conic systems Ax - b \in C_Y,\; x \in C_X, $ where CX and CY are closed, convex cones.

ORDERING OF CONVEX FUZZY SETS : A BRIEF SURVEY AND NEW RESULTS

Concerning with the topics of a fuzzy rnax order, a brief survey on ordering of fuzzy numbers is presented in this article, and we will consider an extension to that of fuzzy sets. An extension of the fuzzy rnax order as a pseudo order is investigated and defined on a class of fuzzy sets on K (n > 1). This order is developed by using a non-empty closed convex cone and characterized by the projection into its dual cone. Especially a structure of the lattice can be illustrated with the class of rectangle-type fuzzy sets.

Constrained Open Mapping Theorems And Applications

Some constrained open mapping theorems are obtained via Ekeland's variational principle. The constraint need only be a closed subset when the mapping is assumed to be Lipschitz, or a closed convex cone when the mapping is assumed to be closed. Generalizations of some previous results of Welsh and others are obtained. Apart from the presence of a constraint and a different method, the differentiability assumptions made are weaker. As applications, two results on the constrained controllability of nonlinear systems are given.

A nonpolyhedral cone of class function inequalities for positive semidefinite matrices

A function f from the symmetric group Sn into R is called a class function if it is constant on each conjugacy class. Let df be the generalized matrix function associated with f, mapping the n-by-n Hermitian matrices to R. For example, if f(σ)=sgn(σ), then df(A)=detA. Let Kn(Kn(R)) denote the closed convex cone of those f for which df(A)⩾0 for all n-by-n positive semidefinite Hermitian (real symmetric) matrices. For n=1,2,3,4 it is known that Kn and Kn(R) are polyhedral and there is a finite set of “test” matrices Tn(Tn(R)) such that f belongs to Kn(Kn(R)) if and only if df(A)⩾0 for each A in Tn(Tn(R)). We show here that K5 and K5(R) are not polyhedral. Thus, for n=5 there is no finite set of “test” matrices sufficient to establish which generalized matrix functions are nonnegative on the positive semidefinite matrices.

Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system

A conic linear system is a system of the form¶P(d): find x that solves b - Ax∈CY, x∈CX,¶ where CX and CY are closed convex cones, and the data for the system is d=(A,b). This system is“well-posed” to the extent that (small) changes in the data (A,b) do not alter the status of the system (the system remains solvable or not). Renegar defined the “distance to ill-posedness”, ρ(d), to be the smallest change in the data Δd=(ΔA,Δb) for which the system P(d+Δd) is “ill-posed”, i.e., d+Δd is in the intersection of the closure of feasible and infeasible instances d’=(A’,b’) of P(·). Renegar also defined the “condition measure” of the data instance d as C(d):=∥d∥/ρ(d), and showed that this measure is a natural extension of the familiar condition measure associated with systems of linear equations. This study presents two categories of results related to ρ(d), the distance to ill-posedness, and C(d), the condition measure of d. The first category of results involves the approximation of ρ(d) as the optimal value of certain mathematical programs. We present ten different mathematical programs each of whose optimal values provides an approximation of ρ(d) to within certain constants, depending on whether P(d) is feasible or not, and where the constants depend on properties of the cones and the norms used. The second category of results involves the existence of certain inscribed and intersecting balls involving the feasible region of P(d) or the feasible region of its alternative system, in the spirit of the ellipsoid algorithm. These results roughly state that the feasible region of P(d) (or its alternative system when P(d) is not feasible) will contain a ball of radius r that is itself no more than a distance R from the origin, where the ratio R/r satisfies R/r≤c1C(d), and such that r≥\(\) and R≤c3C(d), where c1,c2,c3 are constants that depend only on properties of the cones and the norms used. Therefore the condition measure C(d) is a relevant tool in proving the existence of an inscribed ball in the feasible region of P(d) that is not too far from the origin and whose radius is not too small.

An extension of the mixed primal–dual bases algorithm to the case of more constraints than dimensions

Abstract The problem of minimizing a quadratic function with linear inequality constraints is considered with applications to nonparametric regression with shape assumptions. For many problems, the set defined by the constraints is a closed convex cone. The mixed primal–dual bases algorithm (Fraser and Massam, 1989, Scand. J. Statist. 16, 65–75) for regression under inequality constraints finds a least-squares regression estimate over such a cone in a finite number of steps, with the restriction that the number of constraints does not exceed the number of dimensions in the space. Some applications, however, require more constraints than dimensions, and the main purpose of this paper is to extend the algorithm to this more general case. Properties of the constraint cone and its polar cone are presented in the generality necessary in this situation. One surprising result is that the number of generators of the constraint cone can be much larger than the number of generators for the polar cone. Applications are presented for least-squares regression, and regression in which a roughness penalty term is included in the function to minimize.

Bifurcation of periodic solutions to variational inequalities in R κ based on Alexander-Yorke theorem

Variational inequalities $$\begin{gathered} U(t) \in K, \hfill (\dot U(t) - B_\lambda U(t) - G(\lambda ,U(t)),\;\;Z - U(t)) \geqslant 0 for all Z \in K, a .a . t \in (0,T) \hfill \end{gathered} $$ are studied, where K is a closed convex cone in $${\mathbb{R}}^{\kappa } $$ , κ ≥ 3, B λ is a κ × κ matrix, G is a small perturbation, λ a real parameter. The assumptions guaranteeing a Hopf bifurcation at some λ0 for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some λI ≠ λ0. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at λ0 constructed on the basis of the Alexander-Yorke theorem as global bifurcation branches of a certain enlarged system.

A minimax inequality for vector-valued mappings

This paper presents a minimax inequality for vector-valued mappings in Hausdorff topological vector spaces with pointed closed convex cones.

Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions

Let K be a closed convex cone in a Hilbert space (H,〈·,·〉), and let K + be its positive dual cone. The K-eigenvalues of a continuous linear mapping A:H→H are defined via the complementarity system: x∈K, Ax−λx∈K + , 〈x,Ax−λx〉=0. This paper explores two issues related to this concept: existence results, and upper bounds for the number of K-eigenvalues when the cone K is finitely generated. Special attention is devoted to the case of a Pareto cone in a finite dimensional space.

Constrained controllability of some nonlinear systems

We consider the constrained exact local controllability of nonlinear systems and the constrained exact and approximate controllability of semilinear systems. The constraint on the con trol is a time-dependent closed convex cone with possibly empty interior. Some sufficient and necessary conditions are derived by using the controllability of associated linear or nonlinear systems. A result on the solution set of an operator equation with parameter is also given.

Stability of Bifurcating Periodic Solutions of Differential Inequalities in IR3

A bifurcation problem for the inequality , is considered, where K is a closed convex cone in IR3, Aλ a real 3×3 matrix depending continuously on a real parameter λ, G a small perturbation. Small periodic solutions bifurcating at λ0 from the branch of trivial solutions are studied. It is proved that these solutions are stable or they are contained in a certain attracting set Aλ if they correspond to parameters λ for which the trivial solution is unstable. Further, unstable solutions exist among them if they correspond to parameters for which the trivial solution is stable.

Concave iteration semigroups of linear set-valued functions

We consider a concave iteration semigroup of linear continuous set-valued functions defined on a closed convex cone in a separable Banach space. We prove that such an iteration semigroup has a selection which is also an iteration semigroup of linear conti

Structure of optimal trajectories of convex processes

In this paper we study the dynamic properties of optimal trajectories of convex processes where K⊂ R nis a closed convex cone. We show that optimal trajectories of a convex process Gspend most of the time in a small neighborhood of a von Neumann path. The turnpike theorem that we obtain generalizes the result of Rubinov [10] which was established for a convex process where is the cone of the elements of the Euclidean space R nthat have nonnegative coordinates. Also, we show that the turnpike phenomenon is stable under small pertubations of a convex process G.

Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains

The system of linear elasticity is considered in a perforated domain with an ε-periodic structure. External forces nonlinearly depending on the displacements are applied to the surface of the cavities (or channels), while the body is fixed along the outer portion of its boundary. We investigate the asymptotic behavior of solutions to such boundary value problems asε→0 and construct the limit problem, according to the external surface forces and their dependence on the parameter ε. In some cases, this dependence results in the homogenized problem having the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone is described in terms of the functions involved in the nonlinear boundary conditions on the perforated boundary. A homogenization theorem is also proved for some unilateral problems with boundary conditions of Signorini type for the system of elasticity in a perforated domain. We discuss some cases when the homogenized tensor may depend on the functions specifying the boundary conditions.

Directional tests for one-sided alternatives in multivariate models

Consider one-sided testing problems for a multivariate exponential family model. Through conditioning or other considerations, the problem oftentimes reduces to testing a null hypothesis that the natural parameter is a zero vector against the alternative that the natural parameter lies in a closed convex cone $\mathscr{C}$. The problems include testing homogeneity of parameters, testing independence in contingency tables, testing stochastic ordering of distributions and many others. A test methodology is developed that directionalizes the usual test procedures such as likelihood ratio, chi square, Fisher, and so on. The methodology can be applied to families of tests where the family is indexed by a size parameter so as to enable nonrandomized testing by $p$-values. For discrete models, a refined family of tests provides a refined grid for better testing by $p$-values. The tests have essential monotonicity properties that are required for admissibility and for desirable power properties. Two examples are given.

Minimax Approximation of Optical Profiles

The notion of a profile is important in the theory surrounding the finishing of axisymmetric optical surfaces. Mathematically, a profile is a member of the quotient space $C(M)/K$, where $C(M)$ denotes the space of continuous functions defined on a compact subset $M\subset R$, and K is the subspace of constant functions. In this paper we investigate the minimax approximation of a given profile $[f]\in C(M)/K$ by elements of a closed convex cone in $C(M)/K$. We establish the existence of a minimax approximation (uniqueness does not in general hold) and prove two characterization theorems for any such best approximation. One of these theorems is then used as a basis for a ``bisection' algorithm to compute a best approximation corresponding to a particular type of finishing process known as recursive operator controlled finishing.

A Type of Minimax Inequality for Vector-Valued Mappings

In this paper, we establish a type of minimax inequality for vector-valued mappings in Hausdorff topological vector spaces with pointed closed convex cones, which can be viewed as vector versions of the famous Ky Fan minimax inequality.

Increasing iteration semigroups of Jensen set-valued functions

Let S be a closed convex cone with non-empty interior in a real separable Banach space and cc(S) denote the set of all non-empty convex compact subsets of S. A necessary and sufficient condition is given under which a family {f t : t > 0} of set-valued functions f t : S → cc(S) is an increasing iteration semigroup of continuous Jensen set-valued functions.