Euclidean Jordan Algebras Research Articles

The weighted vertical linear complementarity problem on Euclidean Jordan algebra

The weighted vertical linear complementarity problem on Euclidean Jordan algebra

The horizontal tensor complementarity problem

This article explores a new type of nonlinear complementarity problem, namely the horizontal tensor complementarity problem (HTCP), which is a natural extension of the horizontal linear complementarity problem studied in Sznajder [Degree-theoretic analysis of the vertical and horizontal linear complementarity problems [Ph.D. Thesis]. University of Maryland Baltimore County; 1994]. We extend the concepts of R 0 , R and P pairs from a pair of linear transformations given in Chi et al. [The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra. J Global Optim. 2019;73:153–169] to a pair of tensors. When a given pair of tensors has these properties, we use degree-theoretic tools to discuss the existence and boundedness of solutions to the HTCP. Finally, we study a uniqueness result of the solution of the HTCP.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Jordan automorphisms and derivatives of symmetric cones

Hyperbolicity cones, and in particular symmetric cones, are of great interest in optimization. Renegar showed that every hyperbolicity cone has a family of derivative cones that approximate it. Ito and Lourenço found the automorphisms of those derivatives when the original cone is generated by rank-one elements, as symmetric cones happen to be. We show that the derivative automorphisms of a symmetric cone are closely related to the automorphisms of its associated Euclidean Jordan algebra. In the process, we find the automorphism group of the quaternion positive-semidefinite cone and list explicitly the Jordan-automorphisms of the quaternion Hermitian matrices. We also address the path-connectedness of the simple Euclidean Jordan-automorphism groups.

The Jordan algebraic structure of the rotated quadratic cone

In this paper, we look into the rotated quadratic cone and analyze its algebraic structure. We construct an algebra associated with this cone and show that this algebra is a Euclidean Jordan algebra (EJA) with a certain inner product. We also demonstrate some spectral and algebraic characteristics of this EJA. The rotated quadratic cone is then proven to be the cone of squares of the generated EJA. The obtained results can help optimization researchers improve specialized interior-point algorithms for rotated quadratic cone programming based on the generated EJA. Additionally, since it is known that the rotated quadratic cone is a special case of the power cone, another reason for this study may be to open the door to understanding the algebraic structure of the general power cone in the future.

The Schatten $ p $-quasinorm on Euclidean Jordan algebras

<abstract><p>In this article, we proved that a Schatten $ p $-(quasi)norm for $ 0 < p < 1 $, defined on Euclidean Jordan algebras, satisfied a relaxed triangle inequality with an optimal constant $ 2^{\frac{1}{p} - 1} $; hence, it indeed induced a quasinorm. This confirmed the validity of a conjecture raised by Huang, Chen, and Hu.</p></abstract>

Euclidean Jordan Algebras, Symmetric Association Schemes, Strongly Regular Graphs, and Modified Krein Parameters of a Strongly Regular Graph

In this paper, in the environment of Euclidean Jordan algebras, we establish some inequalities over the Krein parameters of a symmetric association scheme and of a strongly regular graph. Next, we define the modified Krein parameters of a strongly regular graph and establish some admissibility conditions over these parameters. Finally, we introduce some relations over the Krein parameters of a strongly regular graph.

Some Inequalities over the Eigenvalues of a Strongly Regular Graph

Let’s consider a primitive strongly regular graph G. In this paper, we establish some inequalities over the spectrum of G in the environment of a real finite dimensional Euclidean Jordan algebra A associated with G recurring to a spectral analysis of some elements of A and recurring to a spectral analysis of the Generalized Krein parameters of G.

A Fiedler-type determinantal inequality in Euclidean Jordan algebras

The Fiedler's determinantal inequality stated by M. Fiedler in [5] (1971) provides bounds for the determinant of the sum of two Hermitian matrices. In this note, we extend the Fiedler's result to Euclidean Jordan algebras. In our proof, we use the commutation principle for variational problems which was recently introduced in these algebras. As an application, we obtain some Minkowski-type determinantal inequalities in Euclidean Jordan algebras.

An Algebraic-Based Primal–Dual Interior-Point Algorithm for Rotated Quadratic Cone Optimization

In rotated quadratic cone programming problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of rotated quadratic cones. In this paper, we introduce the rotated quadratic cone programming problems as a “self-made” class of optimization problems. Based on our own Euclidean Jordan algebra, we present a glimpse of the duality theory associated with these problems and develop a special-purpose primal–dual interior-point algorithm for solving them. The efficiency of the proposed algorithm is shown by providing some numerical examples.

Horofunction Compactifications and Duality

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form B=B_1times cdots times B_r, where B_i is an open Euclidean ball in {mathbb {C}}^{n_i}, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm. For these spaces we show, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the Finsler metric in the tangent space at the basepoint. In each case we give an explicit homeomorphism. For finite dimensional normed spaces the link between the geometry of the horofunction compactification and the dual unit ball was suggested by Kapovich and Leeb, which we confirm for Euclidean Jordan algebras with the spectral norm. Our results also show that this duality phenomenon not only occurs in normed spaces, but also in a variety of noncompact type symmetric spaces with invariant Finsler metrics.

Commutativity, Majorization, and Reduction in Fan–Theobald–von Neumann Systems

A Fan–Theobald–von Neumann system (Gowda in Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ) is a triple $$({{\mathcal {V}}},{{\mathcal {W}}},\lambda )$$ , where $${{\mathcal {V}}}$$ and $${{\mathcal {W}}}$$ are real inner product spaces and $$\lambda :{{\mathcal {V}}}\rightarrow {{\mathcal {W}}}$$ is a norm-preserving map satisfying a Fan–Theobald–von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In Gowda (Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ), we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions. We also introduced the concept of commutativity via the equality in the Fan–Theobald–von Neumann-type inequality. In the present paper, we elaborate on the concept of commutativity and introduce/study automorphisms, majorization, and reduction in Fan–Theobald–von Neumann systems.

Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras

A well-known theorem of Korovkin asserts that if \(\{T_k\}\) is a sequence of positive linear transformations on C[a, b] such that \(T_k(h)\rightarrow h\) (in the sup-norm on C[a, b]) for all \(h\in \{1,\phi ,\phi ^2\}\), where \(\phi (t)=t\) on [a, b], then \(T_k(h)\rightarrow h\) for all \(h\in C[a,b]\). In particular, if T is a positive linear transformation on C[a, b] such that \(T(h)=h\) for all \(h\in \{1,\phi ,\phi ^2\}\), then T is the identity transformation. In this paper, we present some analogs of these results over Euclidean Jordan algebras. We show that if T is a positive linear transformation on a Euclidean Jordan algebra \({{\mathcal {V}}}\) such that \(T(h)=h\) for all \(h\in \{e,p,p^2\}\), where e is the unit element in \({{\mathcal {V}}}\) and p is an element of \({{\mathcal {V}}}\) with distinct eigenvalues, then \(T=T^*=I\) (the identity transformation) on the span of the Jordan frame corresponding to the spectral decomposition of p; consequently, if a positive linear transformation coincides with the identity transformation (more generally, an automorphism of \({{\mathcal {V}}}\)) on a Jordan frame, then it is doubly stochastic. We also present sequential and weak-majorization versions.

Conic Optimization with Spectral Functions on Euclidean Jordan Algebras

Spectral functions on Euclidean Jordan algebras arise frequently in convex optimization models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, that is, proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones—the root-determinant cones and the matrix monotone derivative cones—we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open-source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equivalent extended formulations written using only the cones these solvers support.Funding: This work was supported by Office of Naval Research [Grant N00014-18-1-2079] and the National Science Foundation [Grant OAC-1835443].

Euclidean Jordan Algebras and Some New Inequalities Over the Parameters of a Strongly Regular Graph

Let’s consider a primitive strongly regular graph G and it’s adjacency matrix A. Next we consider the Euclidean subalgebra A of the Euclidean Jordan algebra of real symmetric matrices of order n, with the Jordan product and with the inner product of two matrices as being the usual trace of two matrices. Finally, we make a spectral analysis of an Hadamard series of an element of A to establish some new conditions over the spectrum and the parameters of the primitive strongly regular graph G.

Hadamard product and related inequalities in the Jordan spin algebra

Due to its simple yet elegant structure, the study of an entry-wise product of matrices, called the Hadamard product, has received extensive attention from researchers and has expanded to various disciplines, including Euclidean Jordan algebras. As an ongoing effort to extend this product to Euclidean Jordan algebras, in this article, we propose a Hadamard product in the setting of Jordan spin algebra, , under the scheme of the Peirce decomposition, and show that it preserves the diagonal structure of the elements in the algebra. It is shown that this new product corresponds to the standard Hadamard product of symmetric matrices in the case of . Lastly, we prove that this novel product satisfies an analog of the Schur product theorem as well as the inequalities of Hadamard, Oppenheim, Fiedler, etc.

Condition Number Minimization in Euclidean Jordan Algebras

Condition Number Minimization in Euclidean Jordan Algebras

A wide neighbourhood predictor–corrector infeasible-interior-point algorithm for symmetric cone programming

In this paper, we propose a new predictor–corrector infeasible-interior-point algorithm for symmetric cone programming. Each iterate always follows the usual wide neighbourhood , it does not necessarily stay within it but must stay within the wider neighbourhood . We prove that, besides the predictor step, each corrector step also reduces the duality gap by a rate of , where r is the rank of the associated Euclidean Jordan algebra. Moreover, we improve the theoretical complexity bound of an infeasible-interior-point method. Some numerical results are provided as well.

Rank computation in Euclidean Jordan algebras

Euclidean Jordan algebras are the abstract foundation for symmetric cone optimization. Every element in a Euclidean Jordan algebra has a complete spectral decomposition analogous to and subsuming that of a real symmetric matrix into rank-one projections. This general spectral decomposition stems from the element's likewise-analogous characteristic polynomial whose degree (they all have the same degree) is called the rank of the algebra. As a prerequisite for the spectral decomposition, we derive an algorithm that computes the rank of a Euclidean Jordan algebra and allows us to construct the characteristic polynomials of its elements. The ultimate goal of this work is to support a generic computational framework for solving symmetric cone optimization problems in Jordan-algebraic terms.

Addressing the Feasibility of the Parameters of Strongly Regular Graphs with a MacLaurin Series Approach

In the environment of Euclidean Jordan algebras, we consider a Jordan frame associated to the adjacency matrix of a strongly regular graph and construct a MacLaurin series with an element of the frame. From this series we establish feasibility conditions for the existence of strongly regular graphs.