arXiv subjects

Roland Hildebrand

Publications and source records attributed to Roland Hildebrand.

At least 37 records · Page 2Linked to original sources

arXiv2021

Self-associated three-dimensional cones

For every proper convex cone $K \subset \mathbb R^3$ there exists a unique complete hyperbolic affine 2-sphere with mean curvature $-1$ which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This equivalence relation is generated by the groups $SL(3,\mathbb R)$ and $S^1$, where the former acts by linear transformations of the ambient space, and the latter by multiplication of the cubic holomorphic differential of the affine sphere by unimodular complex constants. The action of $S^1$ generalizes conic duality, which acts by multiplication of the cubic differential by $-1$. We call a cone self-associated if it is linearly isomorphic to all its associated cones, in which case the action of $S^1$ induces (nonlinear) isometries of the corresponding affine sphere. We give a complete classification of the self-associated cones and compute isothermal parametrizations of the corresponding affine spheres. The solutions can be expressed in terms of degenerate Painlevé III transcendents. The boundaries of generic self-associated cones can be represented as conic hulls of vector-valued solutions of a certain third-order linear ordinary differential equation with periodic coefficients, but there exist also cones with polyhedral boundary parts.

math.DG↗

arXiv2020

Optimal step length for the Newton method near the minimum of a self-concordant function

In path-following methods for conic programming knowledge of the performance of the (damped) Newton method at finite distances from the minimizer of a self-concordant function is crucial for the tuning of the parameters of the method. The available bounds on the progress and the step length to be used are based on conservative relations between the Hessians at different points and are hence sub-optimal. In this contribution we use methods of optimal control theory to compute the optimal step length of the Newton method on the class of self-concordant functions, as a function of the initial Newton decrement, and the resulting worst-case decrease of the decrement. The exact bounds are expressed in terms of solutions of ordinary differential equations which cannot be integrated explicitly. We provide approximate numerical and analytic expressions which are accurate enough for use in optimization methods. As an application, the neighbourhood of the central path in which the iterates of path-following methods are required to stay can be enlarged, enabling faster progress along the central path during each iteration and hence fewer iterations to achieve a given accuracy.

math.OC↗

arXiv2020

Graph immersions with parallel cubic form

We consider non-degenerate graph immersions into affine space $\mathbb A^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs $(J,γ)$, where $J$ is an $n$-dimensional real Jordan algebra and $γ$ is a non-degenerate trace form on $J$. Every graph immersion with parallel cubic form can be extended to an affine complete symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra $J$. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form.

math.DG↗

arXiv2020

Natural parameterizations of closed projective plane curves

A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth-Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this parametrization is, however, in general not periodic. We show that by the introduction of an additional scalar parameter $α\leq \frac12$ one can define a projectively invariant $2π$-periodic global parametrization on every simple closed convex sufficiently smooth projective plane curve without inflection points. For non-quadratic curves this parametrization, which we call balanced, is unique up to a shift of the parameter. The curve is an ellipse if and only if $α= \frac12$, and the value of $α$ is a global projective invariant of the curve. The parametrization is equivariant with respect to duality.

math.DG↗

arXiv2017

Extremal copositive matrices with minimal zero supports of cardinality two

Let $A \in {\cal C}^n$ be an extremal copositive matrix with unit diagonal. Then the minimal zeros of $A$ all have supports of cardinality two if and only if the elements of $A$ are all from the set $\{-1,0,1\}$. Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal $\{-1,0,1\}$ unit diagonal matrices characterized by Hoffman and Pereira in 1973.

math.OC↗

arXiv2016

Copositive matrices with circulant zero support set

Let $n \geq 5$ and let $u^1,\dots,u^n$ be nonnegative real $n$-vectors such that the indices of their positive elements form the sets $\{1,2,\dots,n-2\},\{2,3,\dots,n-1\},\dots,\{n,1,\dots,n-3\}$, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms which vanish on the vectors $u^1,\dots,u^n$ is a face of the copositive cone ${\cal C}^n$. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite forms, and study their properties. If the vectors $u^1,\dots,u^n$ and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we say it has minimal circulant zero support set, and otherwise non-minimal circulant zero support set. We show that forms with non-minimal circulant zero support set are always extremal, and forms with minimal circulant zero support sets can be extremal only if $n$ is odd. We construct explicit examples of extremal forms with non-minimal circulant zero support set for any order $n \geq 5$, and examples of extremal forms with minimal circulant zero support set for any odd order $n \geq 5$. The set of all forms with non-minimal circulant zero support set, i.e., defined by different collections $u^1,\dots,u^n$ of zeros, is a submanifold of codimension $2n$, the set of all forms with minimal circulant zero support set a submanifold of codimension $n$.

math.OC↗

arXiv2015

Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side

We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely the chaotic behaviour of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighbourhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us the possibility to calculate the entropy and the Hausdorff dimension of the non-wandering set which appears to have a Cantor-like structure as in Smale's Horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.

math.DS↗

arXiv2015

Spectrahedral cones generated by rank 1 matrices

Let ${\cal S}_+^n \subset {\cal S}^n$ be the cone of positive semi-definite matrices as a subset of the vector space of real symmetric $n \times n$ matrices. The intersection of ${\cal S}_+^n$ with a linear subspace of ${\cal S}^n$ is called a spectrahedral cone. We consider spectrahedral cones $K$ such that every element of $K$ can be represented as a sum of rank 1 matrices in $K$. We shall call such spectrahedral cones rank one generated (ROG). We show that ROG cones which are linearly isomorphic as convex cones are also isomorphic as linear sections of the positive semi-definite matrix cone, which is not the case for general spectrahedral cones. We give many examples of ROG cones and show how to construct new ROG cones from given ones by different procedures. We provide classifications of some subclasses of ROG cones, in particular, we classify all ROG cones for matrix sizes not exceeding 4. Further we prove some results on the structure of ROG cones. We also briefly consider the case of complex or quaternionic matrices. ROG cones are in close relation with the exactness of semi-definite relaxations of quadratically constrained quadratic optimization problems or of relaxations approximating the cone of nonnegative functions in squared functional systems.

math.OC↗

arXiv2014

Minimal zeros of copositive matrices

Let $A$ be an element of the copositive cone ${\cal C}_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set $\mbox{supp}u \subset \{1,\dots,n\}$ corresponding to the positive entries of $u$. A zero $u$ of $A$ is called minimal if there does not exist another zero $v$ of $A$ such that its support $\mbox{supp}v$ is a strict subset of $\mbox{supp}u$. We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone $S_+(n)$ of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix $A$ with respect to $S_+(n)$ in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone ${\cal N}_n$ of entry-wise nonnegative matrices. For $n = 5$ matrices which are irreducible with respect to both $S_+(5)$ and ${\cal N}_5$ are extremal. For $n = 6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.

math.OC↗

arXiv2013

Analytic formulas for complete hyperbolic affine spheres

We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of these cones. The affine spheres are represented by explicit hypersurface immersions into three-dimensional real space. The generic member of the family of immersions is given by elliptic integrals.

math.DG↗

arXiv2013

Hessian potentials with parallel derivatives

Let $U \subset \mathbb A^n$ be an open subset of real affine space. We consider functions $F: U \to \mathbb R$ with non-degenerate Hessian such that the first or the third derivative of $F$ is parallel with respect to the Levi-Civita connection defined by the Hessian metric $F"$. In the former case the solutions are given precisely by the logarithmically homogeneous functions, while the latter case is closely linked to metrised Jordan algebras. Both conditions together are related to unital metrised Jordan algebras. Both conditions combined with convexity provide a local characterization of canonical barriers on symmetric cones.

math.DG↗

arXiv2013

Centro-affine hypersurface immersions with parallel cubic form

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, and Köchers conic omega-domains, which are the maximal connected sets consisting of invertible elements in a real semi-simple Jordan algebra. Every level surface of the omega function in an omega-domain is an affine complete, Euclidean complete proper affine hypersphere with parallel cubic form and with center in the origin. On the other hand, every proper affine hypersphere with parallel cubic form and with center in the origin can be represented as such a level surface. We provide a complete classification of proper affine hyperspheres with parallel cubic form based on the classification of semi-simple real Jordan algebras. Centro-affine hypersurface immersions with parallel cubic form are related to the wider class of real unital Jordan algebras. Every such immersion can be extended to an affine complete one, whose conic hull is the connected component of the unit element in the set of invertible elements in a real unital Jordan algebra. Our approach can be used to study also other classes of hypersurfaces with parallel cubic form.

math.DG↗

arXiv2008

Identification of community structure in networks with convex optimization

We reformulate the problem of modularity maximization over the set of partitions of a network as a conic optimization problem over the completely positive cone, converting it from a combinatorial optimization problem to a convex continuous one. A semidefinite relaxation of this conic program then allows to compute upper bounds on the maximum modularity of the network. Based on the solution of the corresponding semidefinite program, we design a randomized algorithm generating partitions of the network with suboptimal modularities. We apply this algorithm to several benchmark networks, demonstrating that it is competitive in accuracy with the best algorithms previously known. We use our method to provide the first proof of optimality of a partition for a real-world network.

physics.data-an↗

arXiv2007

Combinatorial laplacians and positivity under partial transpose

Density matrices of graphs are combinatorial laplacians normalized to have trace one (Braunstein \emph{et al.} \emph{Phys. Rev. A,} \textbf{73}:1, 012320 (2006)). If the vertices of a graph are arranged as an array, then its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for separability of density matrices of graphs, is equivalent to the PPT-criterion. As such it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (for an alternative proof see Wu, \emph{Phys. Lett. A}\textbf{351} (2006), no. 1-2, 18--22). Finally we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.

cs.CC↗

arXiv2007

Semidefinite descriptions of separable matrix cones

Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.

math.RA↗

arXiv2006

Concurrence of Lorentz-positive maps

Let L_n be the n-dimensional Lorentz cone. A linear map M from R^m to R^n is called Lorentz-positive if M[L_m] is contained in L_n. We extend the notion of concurrence, which was initially introduced to quantify the entanglement of bipartite density matrices, to Lorentz-positive maps and provide an explicite formula for it. This allows us to obtain formulae for the concurrence of arbitrary positive operators taking 2 x 2 complex hermitian matrices as input and consequently of arbitrary bipartite density matrices of rank 2. Namely, let P: H(2) \to H(d) be a positive operator, and let λ_1,...,λ_4 be the generalized eigenvalues of the pencil σ_2(P(X)) - λdet X, in decreasing order, where σ_2 is the second symmetric function of the spectrum. Then the concurrence is given by the expression C(P;X) = 2\sqrt{σ_2(P(X)) - λ_2 det X}. As an application, we compute the concurrences of the density matrices of all graphs with 2 edges. Similar results apply for a function which we call I-fidelity, with the second largest generalized eigenvalue λ_2 replaced by the smallest eigenvalue λ_4.

quant-ph↗

arXiv2006

Separable balls around the maximally mixed state for a 3-qubit system

We obtain a new lower bound on the radius of the largest ball of separable unnormalized states around the identity matrix for a 3-qubit system. This also enables us to improve the corresponding lower bounds for multi-qubit systems. These bounds are approximately 5% better than the previously known ones. As a by-product, we compute the radius of the largest ball that fits into the triple projective tensor product of the unit ball in R^3.

quant-ph↗

arXiv2005

Cones of ball-ball separable elements

Let B_1,B_2 be balls in finite-dimensional real vector spaces E_1,E_2, centered around unit length vectors v_1,v_2 and not containing zero. An element in the tensor product space E_1 \otimes E_2 is called B_1 \otimes B_2-separable if it is contained in the convex conic hull of elements of the form w_1 \otimes w_2, where w_1 \in B_1, w_2 \in B_2. We study the cone formed by the separable elements in E_1 \otimes E_2. We determine the largest faces of this cone via a description of the extreme rays of the dual cone, i.e. the cone of the corresponding positive linear maps. We compute the radius of the largest ball centered around v_1 \otimes v_2 that consists of separable elements. As an application we obtain lower bounds on the radius of the largest ball of separable unnormalized states around the identity matrix for a multi-qubit system. These bounds are approximately 12% better than the best previously known. Our results are extendible to the case where B_1,B_2 are solid ellipsoids.

quant-ph↗