An introduction to a class of matrix cone programming

C. Ding, Defeng Sun, K.-C. Toh

Research output: Journal article publicationJournal articleAcademic researchpeer-review

42 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)141-179
Number of pages39
JournalMathematical Programming
Issue number1-2
Publication statusPublished - 1 Jan 2014
Externally publishedYes


  • Conic optimization
  • Matrix cones
  • Metric projectors

ASJC Scopus subject areas

  • Software
  • General Mathematics


Dive into the research topics of 'An introduction to a class of matrix cone programming'. Together they form a unique fingerprint.

Cite this