Spherical functions on symmetric cones

In this note, we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real Jordan algebra. We use this formula as an inspiration for a similar recursive formula involving the Jack polynomials.

     

Finsler metrics on symmetric cones

Let V be a simple Euclidean Jordan algebra with an associative inner product and let be the corresponding symmetric cone. Let be the compact symmetric space of all primitive idempotents of V. We show that the function s(a,b) defined by is a (the automorphism group of )-invariant complete metric on and it coincides with a natural Finsler distance on We also show that the metric s(a,b) (strictly) contracts any (strict) conformal compression of . Received: 24 May 1999 / in final form: 15 March 1999     

Penalized complementarity functions on symmetric cones

We show that penalized functions of the Fischer–Burmeister and the natural residual functions defined on symmetric cones are complementarity functions. Boundedness of the solution set of a symmetric cone complementarity problem, based on the penalized natural residual function, is proved under monotonicity and strict feasibility. The proof relies on a trace inequality on Euclidean Jordan algebras.     

Applications of geometric means on symmetric cones

Let V be a Euclidean Jordan algebra, and let be the corresponding symmetric cone. The geometric mean of two elements a and b in is defined as a unique solution, which belongs to of the quadratic equation where P is the quadratic representation of V. In this paper, we show that for any a in the sequence of iterate of the function defined by converges to a. As applications, we obtain that the geometric mean of can be represented as a limit of successive iteration of arithmetic means and harmonic means, and we derive the L?wner-Heinz inequality on the symmetric cone Furthermore, we obtain a formula which leads a Golden-Thompson type inequality for the spectral norm on V. Received October 5, 1999 / Revised March 6, 2000 / Published online October 30, 2000     

Bases of convex cones and Borwein's proper efficiency

In this note, we establish some interesting relationships between the existence of Borwein's proper efficient points and the existence of bases for convex ordering cones in normed linear spaces. We show that, if the closed unit ball in a smooth normed space ordered by a convex cone possesses a proper efficient point in the sense of Borwein, then the ordering cone is based. In particular, a convex ordering cone in a reflexive space is based if the closed unit ball possesses a proper efficient point. Conversely, we show that, in any ordered normed space, if the ordering cone has a base, then every weakly compact set possesses a proper efficient point.The research was conducted while the author was working on his PhD Degree under the supervision of Professor J. M. Borwein, whose guidance and valuable suggestions are gratefully appreciated. The author would like to thank two anonymous referees for their constructive comments and suggestions. This research was supported by an NSERC grant and a Mount Saint Vincent University Research Grant.     

Lattice point generating functions and symmetric cones

We show that a recent identity of Beck–Gessel–Lee–Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.     

Lyapunov-type least-squares problems over symmetric cones

The Lyapunov-type least-squares problem over symmetric cone is to find the least-squares solution of the Lyapunov equation with a constraint of symmetric cone in the Euclidean Jordan algebra, and it contains the Lyapunov-type least-squares problem over cone of semidefinite matrices as a special case. In this paper, we first give a detailed analysis for the image of Lyapunov operator in the Euclidean Jordan algebra. Relying on these properties together with some characterizations of symmetric cone, we then establish some necessary and?or sufficient conditions for solution existence of the Lyapunov-type least-squares problem. Finally, we study uniqueness of the least-squares solution.     

The generalized fundamental equation of information on symmetric cones

In this paper we generalize the fundamental equation of information to the symmetric cone domain and find the general solution under the assumption of continuity of the respective functions.     

Craig-Sakamoto's theorem for the Wishart distributions on symmetric cones

A version of Craig-Sakamoto's theorem says essentially that ifX is aN(O,I _n ) Gaussian random variable in ⁿ, and ifA andB are (n, n) symmetric matrices, thenXAX andXBX (or traces ofAXX andBXX) are independent random variables if and only ifAB=0. As observed in 1951, by Ogasawara and Takahashi, this result can be extended to the case whereXX is replaced by a Wishart random variable. Many properties of the ordinary Wishart distributions have recently been extended to the Wishart distributions on the symmetric cone generated by a Euclidean Jordan algebraE. Similarly, we generalize there the version of Craig's theorem given by Ogasawara and Takahashi. We prove that ifa andb are inE and ifW is Wishart distributed, then Tracea.W and Traceb.W are independent if and only ifa.b=0 anda.(b.x)=b.(a.x) for allx inE, where the. indicates Jordan product.Partially supported by NATO grant 92.13.47.     

Differential recursion relations for Laguerre functions on symmetric cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R⁺. This family forms an orthogonal basis for the subspace of L-invariant functions in L²(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L²(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R⁺.     

Nondifferentiable multiobjective symmetric dual programs over cones

Wolfe and Mond-Weir type nondifferentiable multiobjective symmetric dual programs are formulated over arbitrary cones and appropriate duality theorems are established under K-preinvexity/K-convexity/pseudoinvexity assumptions.     

Differential equations and recursion relations for Laguerre functions on symmetric cones

We obtain the differential equation and recurrence relations satisfied by the Laguerre functions on an arbitrary symmetric cone .

     

Fixed points of conformal compressions of symmetric cones

Let be a Euclidean Jordan algebra and let be the associated symmetric cone. In this paper, by the contraction property of conformal compressions of the symmetric cone for the natural Riemannian and Finsler metrics on it, we represent the unique fixed point of a strict conformal compression as a limit of continued fractions on the Euclidean Jordan algebra and as the geometric mean of its images at the origin and the infinity point according to the classical triple and the Ol'shanskii polar decompositions of the compression.     

On hyperbolicity cones associated with elementary symmetric polynomials

Elementary symmetric polynomials can be thought of as derivative polynomials of . Their associated hyperbolicity cones give a natural sequence of relaxations for . We establish a recursive structure for these cones, namely, that the coordinate projections of these cones are themselves hyperbolicity cones associated with elementary symmetric polynomials. As a consequence of this recursion, we give an alternative characterization of these cones, and give an algebraic characterization for one particular dual cone associated with together with its self-concordant barrier functional.     

Smoothing algorithms for complementarity problems over symmetric cones

There recently has been much interest in studying optimization problems over symmetric cones. In this paper, we first investigate a smoothing function in the context of symmetric cones and show that it is coercive under suitable assumptions. We then extend two generic frameworks of smoothing algorithms to solve the complementarity problems over symmetric cones, and prove the proposed algorithms are globally convergent under suitable assumptions. We also give a specific smoothing Newton algorithm which is globally and locally quadratically convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for second-order cone complementarity problems are reported.     

Laguerre functions on symmetric cones and recursion relations in the real case

In this article we derive differential recursion relations for the Laguerre functions on the cone Ω$\Omega$ of positive definite real matrices. The highest weight representations of the group Sp(n,R)$\mathrm{Sp}(n,\mathbb{R})$ play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω+iSym(n,R)$\Omega +i\mathrm{Sym}(n,\mathbb{R})$. We then use the Laplace transform to carry the Lie algebra action over to L²(Ω,d_μν)$L^2(\Omega ,\mathrm{d}{\mu }_{\nu })$. The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R).$\mathfrak{sl}(2,\mathbb{R}).$