Mathematical Programming

, Volume 126, Issue 1, pp 1–29 | Cite as

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

  • Guanghui Lan
  • Zhaosong Lu
  • Renato D. C. Monteiro


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 


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  1. 1.
    Auslender A., Teboulle M.: Interior gradient and proximal methods for convex and conic optimization. SIAM J. Opt. 16, 697–725 (2006)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Burer S., Monteiro R.D.C.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. Ser. B 95, 329–357 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burer S., Monteiro R.D.C.: Local minima and convergence in low-rank semidefinite programming. Math. Program. 103, 427–444 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    d’Aspremont A.: Smooth optimization with approximate gradient. SIAM J. Opt. 19, 1171–1183 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms I. Comprehensive Study in Mathematics, vol. 305. Springer, New York (1993)Google Scholar
  6. 6.
    Hoda, S., Gilpin, A., Peña, J.: A gradient-based approach for computing nash equilibria of large sequential games. Working Paper, Tepper School of Business, Carnegie Mellon University (2006)Google Scholar
  7. 7.
    Korpelevich G.: The extragradient method for finding saddle points and other problems. Eknomika i Matematicheskie Metody 12, 747–756 (1976)MATHGoogle Scholar
  8. 8.
    Lu Z., Nemirovski A., Monteiro R.D.C.: Large-scale semidefinite programming via saddle point mirror-prox algorithm. Math. Program. 109, 211–237 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Nemirovski A.: Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM J. Opt. 15, 229–251 (2005)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Nesterov Y.E.: A method for unconstrained convex minimization problem with the rate of convergence O(1/k 2). Doklady AN SSSR 269, 543–547 (1983) (translated as Sov. Math. Docl.)MathSciNetGoogle Scholar
  11. 11.
    Nesterov Y.E.: Smooth minimization of nonsmooth functions. Math. Program. 103, 127–152 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nesterov Y.E.: Smoothing technique and its applications in semidefinite optimization. Math. Program. 110, 245–259 (2006)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Tütüncü R.H., Toh K.C., Todd M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. 95, 189–217 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Guanghui Lan
    • 1
  • Zhaosong Lu
    • 2
  • Renato D. C. Monteiro
    • 3
  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  3. 3.School of ISyEGeorgia Institute of TechnologyAtlantaUSA

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