- An Introduction to a Class of Matrix Cone Programming Chao Ding(dingchaonus.edu.sg) Defeng Sun(matsundfnus.edu.sg) Kim-Chuan Toh(mattohkcnus.edu.sg) Abstract: In this paper, we define a class of linear conic programming (which we call matrix cone programming or MCP) involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone. MCP has recently found many important applications, for example, in nuclear norm relaxations of affine rank minimization problems. In order to make the defined MCP tractable and meaningful, we must first understand the structure of these epigraphs. So far, only the epigraph of the Frobenius matrix norm, which can be regarded as a second order cone, has been well studied. Here, we take an initial step to study several important properties, including its closed form solution, calm Bouligand-differentiability and strong semismoothness, of the metric projection operator over the epigraph of the $l_1$, $l_\infty$, spectral or operator, and nuclear matrix norm, respectively. These properties make it possible to apply augmented Lagrangian methods, which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming, to this class of MCP problems. The work done in this paper is far from comprehensive. Rather it is intended as a starting point to call for more insightful research on MCP so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come. Keywords: matrix cones, metric projectors, conic optimization Category 1: Convex and Nonsmooth Optimization Category 2: Linear, Cone and Semidefinite Programming Category 3: Nonlinear Optimization Citation: Technical report, Department of Mathematics, National University of Singapore, September/2010 Download: [PDF]Entry Submitted: 09/26/2010Entry Accepted: 09/26/2010Entry Last Modified: 09/26/2010Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.