Closed Convex Cone Research Articles

Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors

A symmetric positive semi-definite (PSD) tensor, which is not sum-of-squares (SOS), is called a PSD non-SOS (PNS) tensor. Is there a fourth order four dimensional PNS Hankel tensor? The answer for this question has both theoretical and practical significance. Under the assumptions that the generating vector v of a Hankel tensor A is symmetric and the fifth element v4 of v is fixed at 1, we show that there are two surfaces M0 and N0 with the elements v2,v6,v1,v3,v5 of v as variables, such that M0≥N0, A is SOS if and only if v0≥M0, and A is PSD if and only if v0≥N0, where v0 is the first element of v. If M0=N0 for a point P=(v2,v6,v1,v3,v5)⊤, there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for such v2,v6,v1,v3,v5. Then, we call such P a PNS-free point. We prove that a 45-degree planar closed convex cone, a segment, a ray and an additional point are PNS-free. Numerical tests check various grid points and report that they are all PNS-free.

Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points

Let $X\subset \mathbb{R}^{n}$ be a set whose interior is connected and dense in $X$, ordered by a closed convex cone $K\subset \mathbb{R}^{n}$ having nonempty interior. Let $T: X\approx X$ be an order-preserving homeomorphism. The following result is proved: Assume the set of periodic points of $T$ is dense in $X$, and $K$ is a polyhedron. Then $T$ is periodic.

On the roots of generalized Wills $\mu$-polynomials

We investigate the roots of a family of geometric polynomials of convex bodies associated to a given measure \mu on the non-negative real line \mathbb R_{\geq 0} , which arise from the so called Wills functional. We study its structure, showing that the set of roots in the upper half-plane is a closed convex cone, containing the non-positive real axis \mathbb R_{\leq0} , and strictly increasing in the dimension, for any measure \mu . Moreover, it is proved that the 'smallest' cone of roots of these \mu -polynomials is the one given by the Steiner polynomial, which provides, for example, additional information about the roots of \mu -polynomials when the dimension is large enough. It will also give necessary geometric conditions for a sequence \{m_i\colon i=0,1,\dots\} to be the moments of a certain measure on \mathbb R_{\geq0} , a question regarding the so called (Stieltjes) moment problem.

Critical angles between two convex cones II. Special cases

The concept of critical angle between two linear subspaces has applications in statistics, numerical linear algebra and other areas. Such concept has been abundantly studied in the literature. Part I of this work is an attempt to build up a theory of critical angles for a pair of closed convex cones. The need of such theory is motivated, among other reasons, by some specific problems arising in regression analysis of cone-constrained data, see Tenenhaus in (Psychometrika 53:503–524, 1988). Angle maximization and/or angle minimization problems involving a pair of convex cones are at the core of our discussion. Such optimization problems are nonconvex in general and their numerical resolution offers a number of challenges. Part II of this work focusses on the practical computation of the maximal angle between specially structured cones.

Critical angles between two convex cones I. General theory

The concept of critical (or principal) angle between two linear subspaces has applications in statistics, numerical linear algebra, and other areas. Such concept has been abundantly studied in the literature, both from a theoretical and computational point of view. Part I of this work is an attempt to build a general theory of critical angles for a pair of closed convex cones. The need of such theory is motivated, among other reasons, by some specific problems arising in regression analysis of cone-constrained data, see Tenenhaus (Psychometrika 53:503–524, 1988). Angle maximization and/or angle minimization problems involving a pair of convex cones are at the core of our discussion. Such optimization problems are nonconvex in general and their numerical resolution offer a number of challenges. Part II of this work focusses on the practical computation of the maximal and/or minimal angle between specially structured cones.

Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space

Abstract In this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using the indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.

CARATHEODORY DIFFERENTIAL EQUATIONS ON CONES

In this paper we prove that almost all, in Baire sense, differential equations with Scorza Dragoni right-hand side, defined on closed convex cone of a Banach space, have unique solution. This solution depends continuously on the right-hand side and on the initial condition. The results are applied to fuzzy differential equations and to differential inclusions.

Extensions of closed convex processes

In this paper we provide some norm preserving extension results for real valued closed convex processes. As a consequence, we characterize afterwards, the elements of best approximation in normed linear spaces by elements of closed convex cones using closed convex processes.

The polar cone of the set of monotone maps

We prove that every element of the polar cone to the closed convex cone of monotone transport maps can be represented as the divergence of a measure field taking values in the positive definite matrices.

On the Theory of Coconvex Bodies

If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams) and in commutative algebra. Such invariants of coconvex sets as volumes, mixed volumes, number of integer points, etc., play an important role. This paper aims at extending various results from the theory of convex bodies to the coconvex setting. These include the Aleksandrov---Fenchel inequality and the Ehrhart duality.

On multidimensional Mandelbrot cascades

Let Z be a random variable with values in a proper closed convex cone , A a random endomorphism of C and N a random integer. We assume that Z, A, N are independent. Given N independent copies of we define a new random variable . Let T be the corresponding transformation on the set of probability measures on C, i.e. T maps the law of Z to the law of . If the matrix has dominant eigenvalue 1, we study existence and properties of fixed points of T having finite non-zero expectation. Existing one-dimensional results concerning T are extended to higher dimensions. In particular we give conditions under which such fixed points of T have multidimensional regular variation in the sense of extreme value theory and we determine the index of regular variation.

Projection onto simplicial cones by a semi-smooth Newton method

In this paper the problem of projection onto a simplicial cone is studied. By using Moreau’s decomposition theorem for projecting onto closed convex cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone is always well defined, and the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented. It is also shown that under a somewhat restrictive assumption, the semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone has finite convergence. Besides, under a mild assumption on the simplicial cone, the generated sequence converges linearly to the solution of the associated system of equations.

Successive Projection Method for Well-Conditioned Matrix Approximation Problems

Matrices are often required to be well-conditioned in a wide variety of areas including signal processing. Problems to find the nearest positive definite matrix or the nearest correlation matrix that simultaneously satisfy the condition number constraint and sign constraints are presented in this paper. Both problems can be regarded as those to find a projection to the intersection of the closed convex cone corresponding to the condition number constraint and the convex polyhedron corresponding to the other constraints. Thus, we can apply a successive projection method, which is a classical algorithm for finding the projection to the intersection of multiple convex sets, to these problems. The numerical results demonstrated that the algorithm effectively solved the problems.

Normal bundles of convex bodies

The normal bundle of an (o-symmetric, smooth and strictly) convex body C in Ed gives rise to a closed convex cone in Ed2, the normal bundle coneNC of C. This article deals with properties of the cone NC and relations between properties of NC and C:The family of normal bundle cones is a meager subset of the space of all closed convex cones in Ed2. To find out whether a cone is a normal bundle cone, a simple criterion is given. The dimension of a normal bundle cone NC is between 12d(d+1) and d2. The lower bound is attained precisely for ellipsoids. As the dimension of NC increases, the ellipsoidal character of the convex body C and the family of linear vector fields which are tangent on the boundary of C decrease. For generic C the dimension of NC is d2. For d=2,3 a complete description of the situation is given. Next, symmetry properties are studied. The cone NC coincides with its polar, its polar transpose, or its transpose, if and only if C is a disc with rotational symmetry of order 4, a Radon disc, a Euclidean ball, or an ellipsoid. A conjecture deals with isometries and linear automorphisms of NC and C. A relation between symmetry properties of a pair of normal bundle cones and orthogonality in two normed spaces is stated. There is a bijection between the family of certain faces of NC and the family of planar shadow boundaries of C with respect to parallel illumination. A conjecture deals with a more general case.

Commutativity of set-valued cosine families

AbstractLet K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F t: t ≥ 0} is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.

A duality between the metric projection onto a convex cone and the metric projection onto its dual in Hilbert spaces

If K and L are mutually dual closed convex cones in a Hilbert space H with the metric projections onto them denoted by PK and PL respectively, then the following two assertions are equivalent: (i) PK is isotone with respect to the order induced by K (i.e. v−u∈K implies PKv−PKu∈K); (ii) PL is subadditive with respect to the order induced by L (i.e. PLu+PLv−PL(u+v)∈L for any u,v∈H). This extends the similar result of A.B. Németh and the author for Euclidean spaces. The extension is essential because the proof of the result for Euclidean spaces is essentially finite dimensional and seemingly cannot be extended for Hilbert spaces. The proof of the result for Hilbert spaces is based on a completely different idea which uses extended lattice operations.

Probabilistic Analysis of the Grassmann Condition Number

We analyze the probability that a random m-dimensional linear subspace of $\mathbb{R}^{n}$ both intersects a regular closed convex cone $C\subseteq\mathbb{R}^{n}$ and lies within distance ? of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number [InlineEquation not available: see fulltext.] for the homogeneous convex feasibility problem ?x?C?0:Ax=0. The Grassmann condition number is a geometric version of Renegar's condition number, which we have introduced recently in Amelunxen and Burgisser (SIAM J. Optim. 22(3):1029---1041 (2012)). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of $A\in\mathbb {R}^{m\times n}$ are chosen i.i.d. standard normal, then for any regular cone C, we have [InlineEquation not available: see fulltext.]. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.

On the Stability of the Motzkin Representation of Closed Convex Sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple \(\left( C,D\right) \) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.

Lipschitz and Hölder continuity results for some functions of cones

We prove the Lipschitz continuity of the maximal angle function on the set of closed convex cones in a Hilbert space. A similar result is obtained for the minimal angle function. On the other hand, we prove that the incenter of a solid cone and the circumcenter of a sharp cone behave in a locally Holderian manner.

Centers of sets with symmetry or cyclicity properties

We introduce an axiomatic formalism for the concept of the center of a set in a Euclidean space. Then we explain how to exploit possible symmetries and possible cyclicities in the set in order to localize its center. Special attention is paid to the determination of centers in cones of matrices. Despite its highly abstract flavor, our work has a strong connection with convex optimization theory. In fact, computing the so-called “incenter” of a solid closed convex cone is a matter of solving a nonsmooth convex optimization program. On the other hand, the concept of the incenter of a solid closed convex cone has a bearing on the complexity analysis and design of algorithms for convex optimization programs under conic constraints.