Self-dual Cones Research Articles

Self-Dual Polyhedral Cones and Their Slack Matrices

Self-Dual Polyhedral Cones and Their Slack Matrices

Algebraic and geometric properties of [formula omitted]-semipositive matrices and [formula omitted]-semipositive cones

Given a proper cone K in the Euclidean space Rn, a square matrix A is said to be K-semipositive if there exists an x∈K such that Ax∈int(K), the topological interior of K. The paper aims to study algebraic and geometrical properties of K-semipositive matrices with special emphasis on the self-dual proper Lorentz cone L+n={x∈Rn:xn≥0,∑i=1n−1xi2≤xn2}. More specifically, we discuss a few necessary and other sufficient algebraic conditions for L+n-semipositive matrices. Also, we provide algebraic characterizations for diagonal and orthogonal L+n-semipositive matrices. Furthermore, given a square matrix A and a proper cone K, geometric properties of the semipositive cone KA,K={x∈K:Ax∈K} and the cone of SA,K={x:Ax∈K} are discussed in terms of their extremals. As L+n is an ellipsoidal cone, at last we find results for the cones KA,L+n and SA,L+n to be ellipsoidal.

Divergences on Symmetric Cones and Medians

We are concerned with divergences on the Cartan–Hadamard Riemannian manifold of symmetric cones, self-dual homogeneous cones in Euclidean spaces, and related optimization problems. We introduce a parameterized version of fidelity on symmetric cones, namely sandwiched quasi-relative entropies, and construct a one-parameter family of divergences based on these entropies. We consider the median minimization problem of finite points over these divergences and establish existence and uniqueness of minimizer. The global linear rate convergence of a gradient projection algorithm for solving the median minimization problem is analyzed based on the derived upper bound of the condition number of the Hessian function.

Positively factorizable maps

We initiate a study of linear maps on Mn(C) that have the property that they factor through a tracial von Neumann algebra (A,τ) via operators Z∈Mn(A) whose entries consist of positive elements from the von-Neumann algebra. These maps often arise in the context of non-local games especially in the synchronous case. We establish a connection with the convex sets in Rn containing self-dual cones and the existence of these maps. The Choi matrix of a map of this kind which factor through an abelian von-Neumann algebra turns out to be a completely positive (CP) matrix. We fully characterize positively factorizable maps whose Choi rank is 2. We also provide some applications of this analysis in finding doubly nonnegative matrices which are not CPSD. A special class of these examples are found from the concept of Unextendible Product Bases in quantum information theory.

Isotonicity of the metric projection with respect to the mutually dual orders and complementarity problems

In this paper, as an extension of the isotone projection cone, we consider the isotonicity of the metric projection operator with respect to the mutually dual orders induced by the cone and its dual cone in Hilbert spaces. We first discuss the relation between the isotonicity of the projection onto the cone and its dual cone. Some sufficient conditions for the isotonicity of the projection with respect to the mutually dual orders onto general cones are then proved. By using the self-dual cone in the hyperplane, we establish some specific isotone projection cones. Some properties and representations of the projection onto these cones are studied. We also prove some heredity and expansivity for the isotonicity of the projection onto these cones. By using the isotonicity characterizations of the projection with respect to the mutually dual orders, some solvability and approximation theorems for the complementarity and conic optimization problems are obtained. In Theorems 5.1–5.4, if the order relation does not satisfy the regularity, to guarantee convergence of isotone Picard iterations, three different order methods are used, respectively. Our results generalize those about the isotone projection cone and methods to establish isotone iterations for complementarity problems.

Gaddum’s test for symmetric cones

A real symmetric matrix A is copositive if $$\left\langle {Ax},{x}\right\rangle \ge 0$$ for all x in the nonnegative orthant. Copositive programming gained fame when Burer showed that hard nonconvex problems can be formulated as completely-positive programs. Alas, the power of copositive programming is offset by its difficulty: simple questions like “is this matrix copositive?” have complicated answers. In 1958, Jerry Gaddum proposed a recursive procedure to check if a given matrix is copositive by solving a series of matrix games. It is easy to implement and conceptually simple. Copositivity generalizes to cones other than the nonnegative orthant. If K is a proper cone, then the linear operator L is copositive on K if $$\left\langle {L \left( {x}\right) },{x}\right\rangle \ge 0$$ for all x in K. Little is known about these operators in general. We extend Gaddum’s test to self-dual and symmetric cones, thereby deducing criteria for copositivity in those settings.

The Cheng-Yau metrics on regular convex cones as harmonic immersions into the symmetric space of positive definite real symmetric matrices

A Riemannian metric $g$ on a domain $\Omega$ in $ \mathbb{R}^n$ defines a map $F_g$ from $(\Omega,g)$ into the symmetric space of positive definite real symmetric $n \times n$ matrices $(\textrm{Sym}^+(n),h)$, where $h$ is the Cheng-Yau metric on $\textrm{Sym}^+(n)$. We show that the map $F_g$ is a harmonic immersion if $\Omega$ is a regular convex cone and $g$ is the Cheng-Yau metric on $\Omega$. We also prove that the map $F_g$ is totally geodesic if $\Omega$ is a homogeneous self-dual regular convex cone and $g$ is the Cheng-Yau metric on $\Omega$.

Formules de Feynmann-Kac pour le modèle de Nelson ultra-violet renormalisée

We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in the non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, then the spectrum of the $N$-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal constant times $g^4N^3$, for all values of the coupling constant $g$. A variational argument also yields an upper bound of the same form for large $g^2N$. We further verify that the semi-groups generated by the ultra-violet renormalized Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a natural self-dual cone, if the Pauli principle is ignored. In another application we discuss continuity properties of elements in the range of the semi-group of the renormalized Nelson Hamiltonian.

Quantum Griffiths Inequalities

We present a general framework of Griffiths inequalities for quantum systems. Our approach is based on operator inequalities associated with self-dual cones and provides a consistent viewpoint of the Griffiths inequality. As examples, we discuss the quantum Ising model, quantum rotor model, Bose--Hubbard model, Hubbard model, and spin-1/2 quantum Heisenberg model. We present a model-independent structure that governs the correlation inequalities.

An order inequality characterizing invariant barycenters on symmetric cones

This paper is concerned with invariant and contractive barycenters on the Wasserstein space of probability measures on metric spaces of non-positive curvature, where the center of gravity, also called the Cartan barycenter, is the canonical barycenter on Hadamard spaces. We establish an order inequality of probability measures on partially ordered symmetric spaces of non-compact type, namely symmetric cones (self-dual homogeneous cones), characterizing the Cartan barycenter among other invariant and contractive barycenters. The derived inequality and partially ordered structures on the probability measure space lead also to significant results on (norm) inequalities including the Ando–Hiai inequality for probability measures on symmetric cones.

Erratum to: CBLIB 2014: a benchmark library for conic mixed-integer and continuous optimization

The below statement was inadvertently missed out during the typesetting process. The correct text is provided below. In Sect. 4.1, the list of certificates a solver could return for conic continuous optimization problems is not complete as claimed, as the paper was published without the following fifth point: certified dual facial reducibility when we are given a feasible point to problem (P), modified such that b and B are fixed to zero, with a zero-valued objective value, c T c + ? C , X ? = 0 (within a tolerance), and non-zero entries of any self-dual cone. This is a facial reduction certificate for (D) showing it to be ill-posed in the sense of Renegar [ 2 ]. This certificate is needed when the problem (P) has an unattained optimal solution or if the objective value can be improved indefinitely even though it has no improving ray (i.e., no certificate of dual infeasibility exists). Both cases are shown to occur in [ 1 ] where an algorithm is furthermore constructed with the property of always returning one of the five certificates. This shows the corrected list of certificates to be complete.

Transition probabilities of normal states determine the Jordan structure of a quantum system

Let Φ : 𝔖(M1) → 𝔖(M2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M1 and M2, respectively. This paper concerns with the situation when Φ preserves (or partially preserves) one of the following three notions of “transition probability” on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an “asymmetric transition probability” P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan ∗-isomorphic, if Φ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states μ and ν, we have PΦ(μ),Φ(ν)=0 if and only if P(μ, ν) = 0, where P stands for PU (respectively, PR or P0). Furthermore, as an extension of Wigner’s theorem, it is shown that there is a Jordan ∗-isomorphism Θ : M2 → M1 satisfying Φ = Θ∗|𝔖(M1) if and only if Φ preserves the “asymmetric transition probability.” This is also equivalent to Φ preserving the Raggio transition probability. Consequently, if Φ preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or “the metric induced by the self-dual cone,” are complete Jordan ∗-invariants for the underlying von Neumann algebras.

Positive definite matrices and the S-divergence

Hermitian positive definite (hpd) matrices form a self-dual convex cone whose interior is a Riemannian manifold of nonpositive curvature. The manifold view comes with a natural distance function but the conic view does not. Thus, drawing motivation from convex optimization we introduce the S-divergence, a distance-like function on the cone of hpd matrices. We study basic properties of the S-divergence and explore its connections to the Riemannian distance. In particular, we show that (i) its square-root is a distance, and (ii) it exhibits numerous nonpositive-curvature-like properties.

Completely mixed linear games on a self-dual cone

Given a finite dimensional real inner product space V with a self-dual cone K, an element e in K∘ (the interior of K), and a linear transformation L on V, the value of the linear game (L,e) is defined byv(L,e):=maxx∈Δ(e)⁡miny∈Δ(e)⁡〈L(x),y〉=miny∈Δ(e)⁡maxx∈Δ(e)⁡〈L(x),y〉, where Δ(e)={x∈K:〈x,e〉=1}. In [5], various properties of a linear game and its value were studied and some classical results of Kaplansky [6] and Raghavan [8] were extended to this general setting. In the present paper, we study how the value and properties change as e varies in K∘. In particular, we study the structure of the set Ω(L) of all e in K∘ for which the game (L,e) is completely mixed and identify certain classes of transformations for which Ω(L) equals K∘. We also describe necessary and sufficient conditions for a game (L,e) to be completely mixed when v(L,e)=0, thereby generalizing a result of Kaplansky [6].

On the game-theoretic value of a linear transformation relative to a self-dual cone

This paper is concerned with a generalization of the concept of value of a (zero-sum) matrix game. Given a finite dimensional real inner product space V with a self-dual cone K, an element e in the interior of K, and a linear transformation L, we define the value of L byv(L):=maxx∈Δ⁡miny∈Δ⁡〈L(x),y〉=miny∈Δ⁡maxx∈Δ⁡〈L(x),y〉, where Δ={x∈K:〈x,e〉=1}. This reduces to the classical value of a square matrix when V=Rn, K=R+n, and e is the vector of ones. In this paper, we extend some classical results of Kaplansky and Raghavan to this general setting. In addition, for a Z-transformation (which is a generalization of a Z-matrix), we relate the value with various properties such as the positive stable property, the S-property, etc. We apply these results to find the values of the Lyapunov transformation LA and the Stein transformation SA on the cone of n×n real symmetric positive semidefinite matrices.

Discrete automorphism groups of convex cones of finite type

AbstractWe investigate subgroups of $\text{SL}(n,\mathbb{Z})$ which preserve an open nondegenerate convex cone in $\mathbb{R}^{n}$ and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on self-dual cones, Weyl groups of certain Kac–Moody algebras, and they do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.

NO DICE THEOREM ON SYMMETRIC CONES

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number \cite{LL1, BK}. A natural question concerned with the S.L.L.N is so called {\it the no dice conjecture}. It is a problem to make a construction of deterministic sequences converging to the least squares mean without any probabilistic arguments. Very recently, Holbrook \cite{Hol} gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook \cite{Hol} and the fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we establish the no dice theorem on general symmetric cones.

Lattice-like operations and isotone projection sets

By using some lattice-like operations which constitute extensions of ones introduced by M.S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect to the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E.A. Ok.

The resolvent average on symmetric cones

Recently Bauschke et al. introduced a new notion of proximal average in the context of convex analysis, and studied this subject systemically in Bauschke et al. (2004, 2007, 2008, 2009) [3–7] from various viewpoints. In addition, this new concept was applied to positive semidefinite matrices under the name of resolvent average, and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix analysis [8] (Bauschke et al., 2010). Inspired by their works and the well-known fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous cone), we study the resolvent average on symmetric cones, and derive corresponding results in a different manner based on a purely Jordan algebraic technique.

Symmetry and Self-Duality in Categories of Probabilistic Models

This note adds to the recent spate of derivations of the probabilistic apparatus of finite-dimensional quantum theory from various axiomatic packages. We offer two different axiomatic packages that lead easily to the Jordan algebraic structure of finite-dimensional quantum theory. The derivation relies on the Koecher-Vinberg Theorem, which sets up an equivalence between order-unit spaces having homogeneous, self-dual cones, and formally real Jordan algebras.