A Characterization of Generalized Monotone Normed Cones

Let C be a cone and consider a quasi-norm p defined on it. We study the structure of the couple （C, p） as a topological space in the case where the function p is also monotone. We characterize when the topology of a quasi-normed cone can be defined by means of a monotone norm. We also define and study the dual cone of a monotone normed cone and the monotone quotient of a general cone. We provide a decomposition theorem which allows us to write a cone as a direct sum of a monotone subcone that is isomorphic to the monotone quotient and other particular subcone.     

Avoiding numerical cancellation in the interior point method for solving semidefinite programs

 The matrix variables in a primal-dual pair of semidefinite programs are getting increasingly ill-conditioned as they approach a complementary solution. Multiplying the primal matrix variable with a vector from the eigenspace of the non-basic part will therefore result in heavy numerical cancellation. This effect is amplified by the scaling operation in interior point methods. A complete example illustrates these numerical issues. In order to avoid numerical problems in interior point methods, we propose to maintain the matrix variables in a Cholesky form. We discuss how the factors of the v-space Cholesky form can be updated after a main iteration of the interior point method with Nesterov-Todd scaling. An analogue for second order cone programming is also developed. Numerical results demonstrate the success of this approach. Received: June 16, 2001 / Accepted: April 5, 2002 Published online: October 9, 2002 Key Words. semidefinite programming – second order cone programming Mathematics Subject Classification (2000): 90C22, 90C20     

Deductive systems of a cone algebra — II: Isomorphism theorem

We prove that there is an isomorphism φ of the lattice of deductive systems of a cone algebra onto the lattice of convex ℓ-subgroups of a lattice ordered group (determined by the cone algebra) such that for any deductive system A of the cone algebra, A is respectively a prime, normal or polar if and only if φ(A) is a prime convex ℓ-subgroup, ℓ-ideal or polar subgroup of the ℓ-group, thus generalizing and extending the result of Rachůnek that the lattice of ideals of a pseudo MV-algebra is isomorphic to the lattice of convex ℓ-subgroups of a unital lattice ordered group.      

The <Emphasis Type="Italic">Q</Emphasis> Method for Symmetric Cone Programming

The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an interior point and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q method for second-order cone programming is accurate. The Q method has also been used to develop a “warm-starting” approach for second-order cone programming. The machinery of Euclidean Jordan algebra, certain subgroups of the automorphism group of symmetric cones, and the exponential map is used in the development of the Newton method. Finally we prove that in the presence of certain non-degeneracies the Jacobian of the Newton system is nonsingular at the optimum. Hence the Q method for symmetric cone programming is accurate and can be used to “warm-start” a slightly perturbed symmetric cone program.     

Computing the radius of pointedness of a convex cone

Let Ξ(H) denote the set of all nonzero closed convex cones in a finite dimensional Hilbert space H. Consider this set equipped with the bounded Pompeiu-Hausdorff metric δ. The collection of all pointed cones forms an open set in the metric space (Ξ(H),δ). One possible way of measuring the degree of pointedness of a cone K is by evaluating the distance from K to the set of all nonpointed cones. The number ρ(K) obtained in this way is called the radius of pointedness of the cone K. The evaluation of this number is, in general, a very cumbersome task. In this note, we derive a simple formula for computing ρ(K), and we propose also a method for constructing a nonpointed cone at minimal distance from K. Our results apply to any cone K whose maximal angle does not exceed 120°. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions

Let (E, τ) be a topological vector space and P a cone in E. We shall define a topology τ _P on E so that (E, τ _P ) is a normable topological vector space and P is a normal cone with normal constant M = 1. Then by using the norm, we shall give some results about common fixed points of two multifunctions on cone metric spaces.     

Commutative extended BCK-algebras

We introduce an extended cone algebra, which generalises a Bosbach’s cone algebra within the framework of extended BCK-algebras and show that every such an algebra is a direct product of an ℓ-group and a cone algebra of Bosbach.     

On the geometry of complete intersection toric varieties

In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to 2n − 2, is exactly 2n − 2 if and only if the cone is a bipyramidal cone, where n > 1 is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones. Received: 4 July 2005     

Quotient normed cones

Given a normed cone (X, p) and a subconeY, we construct and study the quotient normed cone (X/Y,p) generated byY. In particular we characterize the bicompleteness of (X/Y, ‖·‖ ^p ,p) in terms of the bicompleteness of (X, p), and prove that the dual quotient cone ((X/Y)^*, || · ‖·‖^p,p) can be identified as a distinguished subcone of the dual cone (X ^*, || · ||_{p, u}). Furthermore, some parts of the theory are presented in the general setting of the spaceCL(X, Y) of all continuous linear mappings from a normed cone (X, p) to a normed cone (Y, q), extending several well-known results related to open continuous linear mappings between normed linear spaces.     

Extremal barriers on cones with Phragmèn-Lindelöf theorems and other applications

Summary We give a detailed analysis of supersolutions of form rλf(θ) for the classL _α of uniformly elliptic operators in nondivergence form with ellipticity constant α. Fundamental to the analysis is the extremal supersolution rλFλ, α(θ) for each real λ, which is itself a solution of a certain extremal elliptic equation and which remains positive on a cone of wider aperature than any other supersolution of this form. These supersolutions are employed as “barriers” to yeild Phragmen-Lindel?f theorems (λ>0) for unbounded domains contained in a cone, and H?lder continuity (λ>0) and removeable singularity (λ<0) results at boundary points with an exterior cone property. In each case the growth condition 0(r_λ) involved, depending on the aperature of the cone, is best possible. Similar results carry through for operators with singular lower order terms and possibily positive zero order coefficient. This work was partially supported by AFOSR grant number 553–64. Entrata in Redazione il 29 gennaio 1971.     

Invariant Orders on Lie Groups and Coverings of Ordered Homogeneous Spaces

 In this paper we give a characterization of those reductive or solvable connected, not necessarily simply connected, Lie groups which permit a non-degenerate group order. A non-degenerate group ordering on G always defines a pointed generating invariant convex cone W in the Lie algebra of G, but not every such cone arises in this way. The cones that do are called global. To decide whether a given cone is global or not is a difficult problem which for simply connected groups and invariant cones has completely been solved by Gichev.     

On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L ^p (μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L ¹(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.     

Quasi-contractions on a nonnormal cone metric space

Ilić and Rakočević [6] proved a fixed point theorem for quasi-contractive mappings on cone metric spaces when the underlying cone is normal. Recently, Z. Kadelburg, S. Radenović, and V. Rakočević obtained a similar result without using the normality condition but only for a contractive constant λ ∈ (0, 1/2) [8]. In this note, using a new method of proof, we prove this theorem for any contractive constant λ ∈ (0, 1).     

On duality gap in linear conic problems

In their paper “Duality of linear conic problems” Shapiro and Nemirovski considered two possible properties (A) and (B) for dual linear conic problems (P) and (D). The property (A) is “If either (P) or (D) is feasible, then there is no duality gap between (P) and (D)”, while property (B) is “If both (P) and (D) are feasible, then there is no duality gap between (P) and (D) and the optimal values val(P) and val(D) are finite”. They showed that (A) holds if and only if the cone K is polyhedral, and gave some partial results related to (B). Later Shapiro conjectured that (B) holds if and only if all the nontrivial faces of the cone K are polyhedral. In this note we mainly prove that both the “if” and “only if” parts of this conjecture are not true by providing examples of closed convex cone in \mathbbR⁴{\mathbb{R}^{4}} for which the corresponding implications are not valid. Moreover, we give alternative proofs for the results related to (B) established by Shapiro and Nemirovski.     

On an Integral Equation in the Problem of Diffraction of a Plane Wave on a Transparent Circular Cone

The problem of diffraction on a transparent convex cone is studied. A uniqueness theorem is proved for the case where the cone is illuminated by a compact source. For a circular cone, the solution is obtained in the form of Kontorovich-Lebedev integrals and Fourier series expansions. A singular integral equation is deduced for the Fourier coefficients, and its regularization is carried out. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 101–123.     

Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum

A mapping is called isotone if it is monotone increasing with respect to the order defined 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 [1, 2, 3, 4, 5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial [2]. This problem is extended by replacing the projection mapping with a continuous isotone retraction 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 isotone retraction onto the cone whose complement is sharp. This result is used for characterizing a subdual latticial cone by the isotonicity of a generalization of the positive part mapping x ↦ x ⁺. This generalization is achieved by generalizing the infimum for subdual cones. The theoretical results of this paper exhibit fundamental properties of the lattice structure of the space which were not analysed before.     

Invariant Orders on Lie Groups and Coverings of Ordered Homogeneous Spaces

 In this paper we give a characterization of those reductive or solvable connected, not necessarily simply connected, Lie groups which permit a non-degenerate group order. A non-degenerate group ordering on G always defines a pointed generating invariant convex cone W in the Lie algebra of G, but not every such cone arises in this way. The cones that do are called global. To decide whether a given cone is global or not is a difficult problem which for simply connected groups and invariant cones has completely been solved by Gichev. (Received 22 October 1999; in revised form 3 March 2000)     

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes, we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective \mathbb Q{\mathbb Q}-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev’s structure theorem for the cone of nef curves of projective terminal threefolds.     

Multiobjective optimization problems with modified objective functions and cone constraints and applications

In this paper, we consider a differentiable multiobjective optimization problem with generalized cone constraints (for short, MOP). We investigate the relationship between weakly efficient solutions for (MOP) and for the multiobjective optimization problem with the modified objective function and cone constraints [for short, (MOP) _η (x)] and saddle points for the Lagrange function of (MOP) _η (x) involving cone invex functions under some suitable assumptions. We also prove the existence of weakly efficient solutions for (MOP) and saddle points for Lagrange function of (MOP) _η (x) by using the Karush-Kuhn-Tucker type optimality conditions under generalized convexity functions. As an application, we investigate a multiobjective fractional programming problem by using the modified objective function method.     

A Note on Range-Restricted Circuit Covers

 Let Cone(G), Int.Cone(G) and Lat(G) be the cone, the integer cone and the lattice of the incidence vectors of the circuits of graph G. A good range is a set ?⊆ℕ such that Cone (G)∩Lat (G)∩?^E⊆Int.Cone(G) for every graph G(V,E). We give a counterexample to a conjecture of Goddyn [1] stating that ℕ\{1} is a good range. Received: November 26, 1997