Please use this identifier to cite or link to this item: https://doi.org/10.1007/s10107-012-0619-7
 Title: An introduction to a class of matrix cone programming Authors: Ding, C.Sun, D.Toh, K.-C. Keywords: Conic optimizationMatrix conesMetric projectors Issue Date: 2014 Citation: Ding, C., Sun, D., Toh, K.-C. (2014). An introduction to a class of matrix cone programming. Mathematical Programming 144 (1-2) : 141-179. ScholarBank@NUS Repository. https://doi.org/10.1007/s10107-012-0619-7 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 been found to have 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 l1,\,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. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society. Source Title: Mathematical Programming URI: http://scholarbank.nus.edu.sg/handle/10635/102845 ISSN: 14364646 DOI: 10.1007/s10107-012-0619-7 Appears in Collections: Staff Publications

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