Mathematical Programming

, Volume 126, Issue 1, pp 1-29

First online:

Primal-dual first-order methods with \({\mathcal {O}(1/\epsilon)}\) iteration-complexity for cone programming

  • Guanghui LanAffiliated withSchool of Industrial and Systems Engineering, Georgia Institute of Technology Email author 
  • , Zhaosong LuAffiliated withDepartment of Mathematics, Simon Fraser University
  • , Renato D. C. MonteiroAffiliated withSchool of ISyE, Georgia Institute of Technology

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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.


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

Mathematics Subject Classification (2000)

65K05 65K10 90C05 90C22 90C25