Convex Cone Research Articles

Duality for quasiconvex minimization over closed convex cones

Duality for quasiconvex minimization over closed convex cones

Group cohesion under individual regulatory constraints

We consider a group consisting of N business units. We suppose there are regulatory constraints for each unit; more precisely, the net worth of each business unit is required to belong to a set of acceptable risks, assumed to be a convex cone. Because of these requirements, there are less incentives to operate under a group structure, as creating one single business unit, or altering the liability repartition among units, may allow to reduce the required capital. We analyse the possibilities for the group to benefit from a diversification effect and economise on the cost of capital. We define and study the risk measures that allow for any group to achieve the minimal capital, as if it were a single unit, without altering the liability of business units, and despite the individual admissibility constraints. We call these risk measures cohesive risk measures. In the commonotonic case, we show that they are tail expectations but calculated under a different probability.

Correspondence between a new class of generalized cone convexity and higher order duality

Correspondence between a new class of generalized cone convexity and higher order duality

Total positivity in exponential families with application to binary variables

We study exponential families of distributions that are multivariate totally positive of order 2 (MTP2), show that these are convex exponential families, and derive conditions for existence of the MLE. Quadratic exponential familes of MTP2 distributions contain attractive Gaussian graphical models and ferromagnetic Ising models as special examples. We show that these are defined by intersecting the space of canonical parameters with a polyhedral cone whose faces correspond to conditional independence relations. Hence MTP2 serves as an implicit regularizer for quadratic exponential families and leads to sparsity in the estimated graphical model. We prove that the maximum likelihood estimator (MLE) in an MTP2 binary exponential family exists if and only if both of the sign patterns $(1,-1)$ and $(-1,1)$ are represented in the sample for every pair of variables; in particular, this implies that the MLE may exist with $n=d$ observations, in stark contrast to unrestricted binary exponential families where $2^d$ observations are required. Finally, we provide a novel and globally convergent algorithm for computing the MLE for MTP2 Ising models similar to iterative proportional scaling and apply it to the analysis of data from two psychological disorders.

Approximation in weighted spaces of vector functions

In this paper, we present the duality theory for general weighted space of vector functions. We mention that a characterization of the dual of a weighted space of vector functions in the particular case $V \subset C^{+} (X)$ is mentioned by J. B. Prolla in [6]. Also, we extend de Branges lemma in this new setting for convex cones of a weighted spaces of vector functions (Theorem 4.2). Using this theorem, we find various approximations results for weighted spaces of vector functions: Theorems 4.2-4.6 as well as Corollary 4.3. We mention also that a brief version of this paper, in the particular case $V \subset C^{+} (X)$, is presented in [3], Chapter 2, subparagraph 2.5.

Siegel domains over Finsler symmetric cones

Abstract Let Ω be a proper open cone in a real Banach space V. We show that the tube domain V ⊕ i ⁢ Ω {V\oplus i\Omega} over Ω is biholomorphic to a bounded symmetric domain if and only if Ω is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that V is a unital JB-algebra in an equivalent norm and Ω is the interior of { v 2 : v ∈ V } {\{v^{2}:v\in V\}} .

A rigorous numerical formulation for upper bound analysis of reinforced soils using second order cone programming

A rigorous upper bound formulation is established for reinforced soils considering both the tensile rupture of the reinforcement and the relative slippage of the reinforcement-soil interface. To represent its finite tensile strength and negligible compressive strength, a novel strategy is proposed to calculate the plastic dissipation rate of the reinforcement without the incorporation of stress variables. Plastic dissipation rates of the soil, the reinforcement and their interfaces are obtained using only kinematic variables and all flow rules are expressed in terms of linear constraints and second order cones. The solution domain is then discretized using linear strain elements for the soil and constant strain elements for the reinforcement and the interface. Numerical examples are given to show the accuracy of the present formulation. The effect of design parameters such as the tensile strength, the length and the location of the reinforcement is discussed.

Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions

The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the form D(α)zi=azi+ui−v,ui,v∈V, where D(α)f is a Caputo derivative of order α of the function f. Additionally, it is assumed that in the process of the game the evader does not move out of a convex polyhedral cone. The set of admissible controls V is a strictly convex compact and a is a real number. The goal of the group of pursuers is to capture of the evader by no less than m different pursuers (the instants of capture may or may not coincide). The target sets are the origin. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.

On generalized solvability of variable nonlinear integral equations on cones

We study the solvability of multidimensional nonlinear integral equations whose operator is a sum of the Nemytsky and Urysohn type operators, and the desired solution belongs to a cone in a space of summable functions. For a wide class of nonlocal variations of the functions defining the composing operators, we prove that the equation remains solvable in the sense that its residual is arbitrarily small with respect to a suitable norm or a seminorm. The classes of acceptable variations are described by convex cones in corresponding function spaces. We also prove that images of variable operators possess the stability property in the sense that their closures under any of these variations contain a fixed convex cone.

Tight bounds on Lyapunov rank

The Lyapunov rank of a cone is the number of independent equations obtainable from an analogue of the complementary slackness condition in cone programming problems, and more equations are generally thought to be better. Bounding the Lyapunov rank of a proper cone in $$ \mathbb {R}^{n}$$ from above has been an open problem. Gowda and Tao gave an upper bound of $$n^{2} - n$$ that was later improved by Orlitzky and Gowda to $$ \left( {n-1}\right) ^{2}$$ . We settle the matter and show that the Lyapunov rank of $$ \left( {n^{2} - n}\right) /2 + 1$$ corresponding to the Lorentz cone is maximal.

Optimal Inequalities Between Distances in Convex Projective Domains

On any proper convex domain in real projective space there exists a natural Riemannian metric, the Blaschke metric. On the other hand, distances between points can be measured in the Hilbert metric. Using techniques of optimal control, we develop inequalities that provide lower bounds for the Riemannian length of the line segment joining two points of the domain by the Hilbert distance between these points. This strengthens a result of Tholozan. Our estimates are valid for a whole class of Riemannian metrics on convex projective domains, namely those induced by convex non-degenerate centro-affine hypersurface immersions. If the immersions are asymptotic to the boundary of the convex cone over the domain, then we can also upper bound the Riemmanian length. On these classes, and in particular for the Blaschke metric, our inequalities are optimal.

Controllability of a Linear System With Nonnegative Sparse Controls

This paper studies controllability of a discrete-time linear dynamical system using nonnegative and sparse inputs. These constraints on the control input arise naturally in many real-life systems where the external influence on the system is unidirectional, and activating each input node adds to the cost of control. We derive the necessary and sufficient conditions for controllability of the system, without imposing any constraints on the system matrices. Unlike the well-known Kalman rank based controllability criteria, the conditions presented in this paper can be verified in polynomial time, and the verification complexity is independent of the sparsity level. The proof of the result is based on the analytical tools concerning the properties of a convex cone. Our results also provide a closed-form expression for the minimum number of control nodes to be activated at every time instant to ensure controllability of the system using positive controls.

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Let V \mathtt {V} be a standard subspace in the complex Hilbert space H \mathcal {H} and G G be a finite dimensional Lie group of unitary and antiunitary operators on H \mathcal {H} containing the modular group ( Δ V i t ) t ∈ R (\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}} of V \mathtt {V} and the corresponding modular conjugation J V J_{\mathtt {V}} . We study the semigroup \[ S V = { g ∈ G ∩ U ⁡ ( H ) : g V ⊆ V } S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] and determine its Lie wedge L ⁡ ( S V ) = { x ∈ g : exp ⁡ ( R + x ) ⊆ S V } \operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak {g} \colon \exp (\mathbb {R}_+ x) \subseteq S_{\mathtt {V}}\} , i.e., the generators of its one-parameter subsemigroups in the Lie algebra g \mathfrak {g} of G G . The semigroup S V S_{\mathtt {V}} is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form G exp ⁡ ( i C ) G \exp (iC) , where C ⊆ g C \subseteq \mathfrak {g} is an Ad ⁡ ( G ) \operatorname {Ad}(G) -invariant closed convex cone. Our main results assert that the Lie wedge L ⁡ ( S V ) \operatorname {\textbf {L}}(S_{\mathtt {V}}) spans a 3 3 -graded Lie subalgebra in which it can be described explicitly in terms of the involution τ \tau of g \mathfrak {g} induced by J V J_{\mathtt {V}} , the generator h ∈ g τ h \in \mathfrak {g}^\tau of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup S V S_{\mathtt {V}} itself.

De Branges Type Lemma and Approximation in Weighted Spaces

The purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

A reflective methane concentration sensor based on biconvex cone photonic crystal fiber

A methane sensor based on photonic crystal fiber Mach-Zehnder interference is proposed. The sensor is a reflective sensor made by coating a section of photonic crystal fiber cladding with cryptophane molecule A/E sensitive film and fusing in the middle of single-mode fiber, forming two convex cones at the node and silver plating on the end face. The sensing principle is analyzed based on the mode interference theory of photonic crystal fiber, and the fabrication method of sensing film and sensing probe is introduced. The methane concentration response results when the sensing film is cryptophane molecule A or cryptophane molecule E is studied by experiments. The sensitivity of the sensor with cryptophane molecule A is 1.069 nm/% when the methane concentration is 0–5%. When the sensitive film is cryptophane molecule E, the sensitivity of the sensor is 1.272 nm/%. Due to its high sensitivity, simple structure and low cost, the sensor has a good prospect in various application fields.

Cdk5 and GSK3\u03b2 inhibit fast endophilin-mediated endocytosis

Endocytosis mediates the cellular uptake of micronutrients and cell surface proteins. Fast Endophilin-mediated endocytosis, FEME, is not constitutively active but triggered upon receptor activation. High levels of growth factors induce spontaneous FEME, which can be suppressed upon serum starvation. This suggested a role for protein kinases in this growth factor receptor-mediated regulation. Using chemical and genetic inhibition, we find that Cdk5 and GSK3β are negative regulators of FEME. They antagonize the binding of Endophilin to Dynamin-1 and to CRMP4, a Plexin A1 adaptor. This control is required for proper axon elongation, branching and growth cone formation in hippocampal neurons. The kinases also block the recruitment of Dynein onto FEME carriers by Bin1. As GSK3β binds to Endophilin, it imposes a local regulation of FEME. Thus, Cdk5 and GSK3β are key regulators of FEME, licensing cells for rapid uptake by the pathway only when their activity is low.

On the sufficiency of [formula omitted]-positivity for truncated compactly supported generalized moment problems

Given a compact set K and a finite set of continuous basis functions, the truncated generalized K-moment problem asks for a characterization of all sequences that can be obtained as moments, with respect to the basis functions, of some nonnegative measure with support in K. Under a condition that a certain convex cone is nonempty, the moment sequences can be characterized as being the elements in the dual cone of the closure. This dual cone is also known as the set of all sequences for which the corresponding Riesz functional is K-positive. Here, we give a short, alternative proof of this statement, based on convex optimization and duality. We then present two examples. The first example shows that if the nonemptiness condition is removed, then K-positivity is in general no longer a sufficient condition for a sequence to be a moment sequence. Nevertheless, the second example shows that there are moment problems where the convex cone is empty, but for which K-positivity is still a necessary and sufficient condition for the existence of a representing measure.

Rational polyhedral outer-approximations of the second-order cone

It is well-known that the second-order cone can be outer-approximated to an arbitrary accuracy ϵ by a polyhedral cone of compact size defined by irrational data. In this paper, we propose two rational polyhedral outer-approximations of compact size retaining the same guaranteed accuracy ϵ. The first outer-approximation has the same size as the optimal but irrational outer-approximation from the literature. In this case, we provide a practical approach to obtain such an approximation defined by the smallest integer coefficients possible, which requires solving a few, small-size integer quadratic programs. The second outer-approximation has a size larger than the optimal irrational outer-approximation by a linear additive factor in the dimension of the second-order cone. However, in this case, the construction is explicit, and it is possible to derive an upper bound on the largest coefficient, which is sublinear in ϵ and logarithmic in the dimension. We also propose a third outer-approximation, which yields the best possible approximation accuracy given an upper bound on the size of its coefficients. Finally, we discuss two theoretical applications in which having a rational polyhedral outer-approximation is crucial, and run some experiments which explore the benefits of the formulations proposed in this paper from a computational perspective.

On angles between convex sets in Hilbert spaces

The notion of the angle between two subspaces has a long history, dating back to Friedrichs's work in 1937 and Dixmier's work on the minimal angle in 1949. In 2006, Deutsch and Hundal studied extensions to convex sets in order to analyze convergence rates for the cyclic projections algorithm. In this work, we characterize the positivity of the minimal angle between two convex cones. We show the existence of, and necessary conditions for, optimal solutions of minimal angle problems associated with two convex subsets as well. Moreover, we generalize a result by Deutsch on minimal angles from linear subspaces to cones. This generalization yields sufficient conditions for the closedness of the sum of two closed convex cones. This also relates to conditions proposed by Beutner and by Seeger and Sossa. Furthermore, we investigate the relation between the intersection of two cones (at least one of which is nonlinear) and the intersection of the polar and dual cones of the underlying cones. It turns out that the two angles involved cannot be positive simultaneously. Various examples illustrate the sharpness of our results.

Structural Analysis of Composite Conical Convex-Concave Plate (CCCP) Using VAM-Based Equivalent Model

Compared with the ordinary foundation plate, the composite conical convex-concave plate (CCCP) has obvious anisotropic characteristics, and there is less research on the relationship between its mechanical properties and structural parameters. In this article, a numerical model for the equivalent stiffness of a typical unit cell with conical convex is established by using the variational asymptotic method. Then, the 3D finite element model (3D-FEM) of CCCP is transformed into 2D equivalent plate model (2D-EPM) with the effective plate properties obtained from the constitutive analysis of unit cell. The accuracy of 2D-EPM is verified by comparing with the displacement, natural frequencies, and buckling results from 3D-FEM under different boundary conditions. Then, the influence of geometric parameters and layup configurations on the effective performances of CCCP are investigated. Finally, the buckling loads and natural frequencies of bidirectional CCCP are compared with those of CCCP by using the present model. The present model is particularly useful in the early design stage of CCCP where many design trade-offs need to be made over a vast design space in terms of material selection, ply angles, and geometric parameters.