Mathematical Programming

, Volume 144, Issue 1–2, pp 141–179 | Cite as

An introduction to a class of matrix cone programming

  • Chao Ding
  • Defeng SunEmail author
  • Kim-Chuan Toh
Full Length Paper Series A


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 \(l_1,\,l_\infty \), 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.


Matrix cones Metric projectors Conic optimization 

Mathematics Subject Classification (2000)

65K05 90C25 90C30 



We wish to thank the anonymous referees and the Associate Editor for helpful comments that led to an improved version of the original submission.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.National Center for Mathematics and Interdisciplinary SciencesChinese Academy of SciencesBeijingP. R. China
  2. 2.Department of Mathematics and Risk Management InstituteNational University of SingaporeSingaporeRepublic of Singapore
  3. 3.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore

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