This paper presents the new concept of second-order cone approximations for convex conic programming. Given any open convex cone K, a logarithmically homogeneous self-concordant barrier for K and any positive real number r ≤ 1, we associate, with each direction x ∈ K, a second-order cone K̂r(x) containing K. We show that K is the intersection of the second-order cones K̂r(x), as x ranges through ...

We give explicit polynomial-sized (in n and k) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k in n variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we const...

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.

We compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions, in terms of integrals related to Mehta’s integral. Several applications for the probabilistic analysis of semidefinite programming are given. AMS subject classifications: 15B48, 52A55, 53C65, 60D05, 90C22

In the present paper, a newly combined higher-order non-differentiable symmetric duality in scalar-objective programming over arbitrary cones is formulated. literature we have discussed primal-dual results with cones, while this article, derived result one model cones. The theorems of are for these problems under ?-pseudoinvexity/?-invexity/C-pseudoconvexity/C-convexity speculations