We study analyticity, differentiability, and semismoothness of Löwner’s operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan a...

We introduce and study the GIT cone of M0,n, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients (P)//SL(2). As one application, we prove unconditionally that the log canonical models of M0,n with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson.

An important property of the Kalman filter is that the underlying Riccati flow is a contraction for the natural metric of the cone of symmetric positive definite matrices. The present paper studies the geometry of a low-rank version of the Kalman filter. The underlying Riccati flow evolves on the manifold of fixed rank symmetric positive semidefinite matrices. Contraction properties of the low-...

The properties of the barrier F (x) = −log(det(x)), defined over the cone of squares of a Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, in a more direct and suitable way th...

We develop a natural variant of Dikin’s affine-scaling method, first for semidefinite programming and then for hyperbolic programming in general. We match the best complexity bounds known for interior-point methods. All previous polynomial-time affine-scaling algorithms have been for conic optimization problems in which the underlying cone is symmetric. Hyperbolicity cones, however, need not be...

We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in [5] depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang [17], where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over t...

Abstract. We introduce two pairs of nondifferentiable multiobjective second order symmetric dual problems with cone constraints over arbitrary closed convex cones, which is different from the one proposed by Mishra and Lai [12]. Under suitable second order pseudo-invexity assumptions we establish weak, strong and converse duality theorems as well as self-duality relations. Our symmetric duality...

Suppose x̄ and s̄ lie in the interiors of a cone K and its dual K∗ respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x̄ and s̄ and incribed in K and K∗ respectively is maximized. Here the balls are defined using the two dual norms. We provide a solution when the cones are symmetric, that is self-dual and homogeneous. This prov...