Motivated by the similarities between the properties of Z-matrices on Rn + and Lyapunov and Stein transformations on the semidefinite cone S+, we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Final...

There are two optical processes that control the retinal image sampled by the photoreceptor array: aberrations of the ocular optics and cone directionality (Stiles-Crawford effect). The shape of wavefront aberration and Stiles-Crawford functions are known to vary markedly across subjects. In this study we investigate in twelve subjects the symmetry between right and left eyes of wavefront aberr...

The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Many of studies done on the SCCP can be categorized into the three research themes, interior point methods for the SCCP, merit or smoothing function methods for the SCCP, and various properties of the SCCP. In thi...

In this paper, a new pair of Mond-Weir type multiobjective second-order symmetric dual models with cone constraints is formulated in which the objective function is optimised with respect to an arbitrary closed convex cone. Usual duality relations are further established under K-η-bonvexity/second-order symmetric dual K-H-convexity assumptions. A nontrivial example has also been illustrated to ...

denote the kth elementary symmetric function, and let Γk denote the connected component of {λ ∈ R : σk(λ) > 0} containing the positive cone {λ ∈ R : λ1 > 0, · · · , λn > 0}. It is well known that Γk = {λ ∈ R : σl(λ) > 0, 1 ≤ l ≤ k}. Let S denote the set of n× n real symmetric matrices. For any A ∈ S we denote by λ(A) the eigenvalues of A. Throughout this note we will assume that Γ ⊂ R is an ope...

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to the symmetric cone constraint. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithm for convex programming with nonnegative orthant cone constraint, we sh...

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we sho...

‎A full Nesterov-Todd (NT) step infeasible interior-point algorithm‎ ‎is proposed for solving monotone linear complementarity problems‎ ‎over symmetric cones by using Euclidean Jordan algebra‎. ‎Two types of‎ ‎full NT-steps are used‎, ‎feasibility steps and centering steps‎. ‎The‎ ‎algorithm starts from strictly feasible iterates of a perturbed‎ ‎problem‎, ‎and, using the central path and feasi...