Isotone retraction cones in Hilbert spaces

A mapping is called isotone if it is monotone increasing with respect to the order induced by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in Isac and Németh (1986, 1990, 1992) [1], [2], [3], [4] and [5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial (Isac (1990) [2]). This problem is extended by replacing the projection mapping with continuous retractions onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous retraction onto the cone whose complement is sharp. Several particular cases are considered and examples are given.     

An infinite class of convex tangent cones

Since the early 1970's, there have been many papers devoted to tangent cones and their applications to optimization. Much of the debate over which tangent cone is best has centered on the properties of Clarke's tangent cone and whether other cones have these properties. In this paper, it is shown that there are an infinite number of tangent cones with some of the nicest properties of Clarke's cone. These properties are convexity, multiple characterizations, and proximal normal formulas. The nature of these cones indicates that the two extremes of this family of cones, the cone of Clarke and the B-tangent cone or the cone of Michel and Penot, warrant further study. The relationship between these new cones and the differentiability of functions is also considered.     

Ordering Epic R-fields

We are concerned with the following problem. Given a ring R and an epic R-field K, under what conditions can K be fully ordered? Epic R-fields can be constructed in terms of matrices over R; this makes it natural to consider matrix cones over R rather than ordinary cones of elements of K. Essentially, a matrix cone over R, associated with a given ordering of K, consists of all square matrices which either become singular or have positive Dieudonné determinant over K. We give necessary and sufficient conditions in terms of matrix cones for (i) an epic R-field to be orderable, (ii) a full order on R to be extendable to a field of fractions of R and (iii) for such an extension to be unique.     

Cones with a mapping cone symmetry in the finite-dimensional case

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio?kowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.     

Characterization of Solid Cones

The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Lattice-Ordered Matrix Rings Over the Integers

We show that the only compatible lattice order on a matrix ring over the integers for which the identity matrix is positive is (up to isomorphism) the usual, entrywise, lattice order. We also find a condition that guarantees that the only compatible lattice order on a matrix ring over the integers is formed by multiplying the positive cone of the usual, entrywise, lattice order by a matrix with positive entries. Using this condition, we show that such orders are the only compatible ones in the two-by-two case.     

On the Density of Positive Proper Efficient Points in a Normed Space

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S ⁺, we establish some density results of positive weak* efficient elements of A in E(A, S ⁺).     

A combinatorial proof of a fixed point property

A class of finite simplicial complexes, called pseudo cones, is developed that has a number of useful combinatorial properties. A partially ordered set is a pseudo cone if its order complex is a pseudo cone. Pseudo cones can be constructed from other pseudo cones in a number of ways. Pseudo cone ordered sets include finite dismantlable ordered sets and finite truncated noncomplemented lattices. The main result of the paper is a combinatorial proof of the fixed simplex property for finite pseudo cones in which a combinatorial structure is constructed that relates fixed simplices to one another. This gives combinatorial proofs of some well known non-constructive results in the fixed point theory of finite partially ordered sets.     

Cones Admitting Strictly Positive Functionals and Scalarization of Some Vector Optimization Problems

This paper is concerned with cones admitting strictly positive functionals and scalarization methods in multiobjective optimization. Assuming that the ordering cone admits strictly positive functionals or possesses a base in normed spaces or is a supernormal cone in a Banach space, we give scalar and scalar proper representations for vector optimization problems with convex and naturally quasiconvex data.     

Generalized tangent cone and an optimization problem in a normed space

Most abstract multiplier rules in the literature are based on the tangential approximation at a point to some set in a Banach space. The present paper is concerned with the study of a generalized tangent cone, which is a tangential approximation to that set at a common point of two sets. The new notion of tangent cone generalizes previous concepts of tangent cones. This generalized tangent cone is used to characterize the optimality conditions for a simultaneous maximization and minimization problem. The paper is of theoretical character; practical applications are not found so far.     

Extremality of stochastic matrices in nonusual cones

The set of doubly stochastic matrices can be regarded as a set in a cone. Thus, one can perturb the cone in different ways, in such a way that it is important to characterize the new extremals. In addition, we study some special families of cones obtaining the corresponding extremals.This paper has been supported by a grant from CONICET, Buenos Aires, Argentina.     

New proof and some generalizations of the minimum principle in optimal control

An optimal control problem is reduced to the finite-dimensional problem of minimizing the terminal payoff over the intersection of the target set with the reachable set. The pointwise Pontryagin minimum principle is derived from two simple preliminary results: the first states that the intersection of two inseparable derived cones at a common point of two given sets is contained in the quasitangent cone (hence, in the contingent cone) to their intersection; the second identifies a derived cone to the reachable set. The standard variants of the minimum principle are easily generalized to problems defined by non-differentiable terminal payoffs on arbitrary target sets.     

Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces

In this paper, as the extension of the isotonicity of the metric projection, the isotonicity characterizations with respect to two arbitrary order relations induced by cones of the metric projection operator are studied in Hilbert spaces, when one cone is a subdual cone and some relations between the two orders hold. Moreover, if the metric projection is not isotone in the whole space, we prove that the metric projection is isotone in some domains in both Hilbert lattices and Hilbert quasi-lattices. By using the isotonicity characterizations with respect to two arbitrary order relations of the metric projection, some solvability and approximation theorems for the complementarity problems are obtained. Our results generalize and improve various recent results in the field of study.     

Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476]. We shall prove that there are no normal cones with normal constant M<1 and for each k>1 there are cones with normal constant M>k. Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], we obtain generalizations of the results.     

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.     

Generalized linear complementarity problems

It has been shown by Lemke that if a matrix is copositive plus on ⁿ , then feasibility of the corresponding linear complementarity problem implies solvability. In this article we show, under suitable conditions, that feasibility of ageneralized linear complementarity problem (i.e., defined over a more general closed convex cone in a real Hilbert space) implies solvability whenever the operator is copositive plus on that cone. We show that among all closed convex cones in a finite dimensional real Hilbert Space, polyhedral cones are theonly ones with the property that every copositive plus, feasible GLCP is solvable. We also prove a perturbation result for generalized linear complementarity problems.This research has been partially supported by the Air Force Office of Scientific Research under grants #AFOSR-82-0271 and #AFOSR-87-0350.     

Steady-state solutions of a reaction–diffusion system arising from intraguild predation and internal storage

Intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat. At first, we established the sharp a priori estimates for nonnegative solutions of the system, which assure that all of nonnegative solutions belong to a special cone. The selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions. Secondly, existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone. It turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive.     

Supermodular Functions on Finite Lattices

The supermodular order on multivariate distributions has many applications in financial and actuarial mathematics. In the particular case of finite, discrete distributions, we generalize the order to distributions on finite lattices. In this setting, we focus on the generating cone of supermodular functions because the extreme rays of that cone (modulo the modular functions) can be used as test functions to determine whether two random variables are ordered under the supermodular order. We completely determine the extreme supermodular functions in some special cases.     

Freef-modules

Let R be a directed ring whose cone of positive elements is strict and satisfies Ore's condition. The main result: there exists a freef-module over an o-module_RM with cone p_m if and only if p_M is half-closed (this generalizes Vainberg's theorem for ordered Abelian groups). In this connection various characterizations off-modules with strict cones are given.