We introduce a hybrid projection-localization method for solving large convex cone programs. The method interleaves a series of projections onto localization sets with the classical alternating projections method for convex feasibility problems; the problem is made amenable to the method by reducing it to a convex feasibility problem composed of a subspace and a cone via its homogeneous self-du...

In this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $X$,  where $X$ is a $C$-distinguished topological space. Then, we show that their dual spaces can be identified in a natural way with certain spaces of Radon measures.

A new endmember extraction method has been developed that is based on a convex cone model for representing vector data. The endmembers are selected directly from the data set. The algorithm for finding the endmembers is sequential: the convex cone model starts with a single endmember and increases incrementally in dimension. Abundance maps are simultaneously generated and updated at each step. ...

Let Ω ⊆ Rn be a compact convex set and x be a point in the exterior of Ω. The aperture angle of x relative to Ω is defined as the maximal angle of the smallest closed convex cone that contains Ω − x. This note provides an explicit formula, based on eigenvalues of symmetric matrices, for the aperture angle of a point relative to an ellipsoid.

We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms whose iteration complexity we analyse are so-called short-step algorithms. Our iteration complexity bounds match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming...

Let Ω ⊆ Rn be a compact convex set and x be a point in the exterior of Ω. The aperture angle of x relative to Ω is defined as the maximal angle of the smallest closed convex cone that contains Ω − x. This note provides an explicit formula, based on eigenvalues of symmetric matrices, for the aperture angle of a point relative to an ellipsoid.

We discuss some extremality issues concerning the circumradius, the inradius, and the condition number of a closed convex cone in Rn. The condition number refers to the ratio between the circumradius and the inradius. We also study the eccentricity of a closed convex cone, which is a coefficient that measures to which extent the circumcenter differs from the incenter.

We express the probability distribution of the solution of a random (standard Gaussian) instance of a convex cone program in terms of the intrinsic volumes and curvature measures of the reference cone. We then 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 Meht...