Primal-dual first-order methods with O(1/∈) iteration-complexity for cone programming

Guanghui Lan, Zhaosong Lu, Renato D C Monteiro

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103 Scopus citations


In this paper we consider the general cone programming problem, and propose primal-dual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov's optimal method (Nesterov in Doklady AN SSSR 269:543-547, 1983; Math Program 103:127-152, 2005), Nesterov's smooth approximation scheme (Nesterov in Math Program 103:127-152, 2005), and Nemirovski's prox-method (Nemirovski in SIAM J Opt 15:229-251, 2005), and propose a variant of Nesterov's optimal method which has outperformed the latter one in our computational experiments. We also derive iteration-complexity bounds for these first-order methods applied to the proposed primal-dual reformulations of the cone programming problem. The performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances. We also compare the approach based on the variant of Nesterov's optimal method with the low-rank method proposed by Burer and Monteiro (Math Program Ser B 95:329-357, 2003; Math Program 103:427-444, 2005) for solving a set of randomly generated SDP instances.

Original languageEnglish (US)
Pages (from-to)1-29
Number of pages29
JournalMathematical Programming
Issue number1
StatePublished - Jan 2011
Externally publishedYes


  • Cone programming
  • Linear programming
  • Nonsmooth method
  • Primal-dual first-order methods
  • Prox-method
  • Semidefinite programming
  • Smooth optimal method


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