Primaldual firstorder methods with \({\mathcal {O}(1/\epsilon)}\) iterationcomplexity for cone programming
 Guanghui Lan,
 Zhaosong Lu,
 Renato D. C. Monteiro
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
In this paper we consider the general cone programming problem, and propose primaldual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss firstorder 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 proxmethod (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 iterationcomplexity bounds for these firstorder methods applied to the proposed primaldual 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 lowrank 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.
 Auslender, A., Teboulle, M. (2006) Interior gradient and proximal methods for convex and conic optimization. SIAM J. Opt. 16: pp. 697725 CrossRef
 Burer, S., Monteiro, R.D.C. (2003) A nonlinear programming algorithm for solving semidefinite programs via lowrank factorization. Math. Program. Ser. B 95: pp. 329357 CrossRef
 Burer, S., Monteiro, R.D.C. (2005) Local minima and convergence in lowrank semidefinite programming. Math. Program. 103: pp. 427444 CrossRef
 d’Aspremont, A. (2008) Smooth optimization with approximate gradient. SIAM J. Opt. 19: pp. 11711183 CrossRef
 HiriartUrruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms I. Comprehensive Study in Mathematics, vol. 305. Springer, New York (1993)
 Hoda, S., Gilpin, A., Peña, J.: A gradientbased approach for computing nash equilibria of large sequential games. Working Paper, Tepper School of Business, Carnegie Mellon University (2006)
 Korpelevich, G. (1976) The extragradient method for finding saddle points and other problems. Eknomika i Matematicheskie Metody 12: pp. 747756
 Lu, Z., Nemirovski, A., Monteiro, R.D.C. (2007) Largescale semidefinite programming via saddle point mirrorprox algorithm. Math. Program. 109: pp. 211237 CrossRef
 Nemirovski, A. (2005) Proxmethod with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convexconcave saddle point problems. SIAM J. Opt. 15: pp. 229251 CrossRef
 Nesterov, Y.E. (1983) A method for unconstrained convex minimization problem with the rate of convergence O(1/k 2). Doklady AN SSSR 269: pp. 543547
 Nesterov, Y.E. (2005) Smooth minimization of nonsmooth functions. Math. Program. 103: pp. 127152 CrossRef
 Nesterov, Y.E. (2006) Smoothing technique and its applications in semidefinite optimization. Math. Program. 110: pp. 245259 CrossRef
 Tütüncü, R.H., Toh, K.C., Todd, M.J. (2003) Solving semidefinitequadraticlinear programs using SDPT3. Math. Program. 95: pp. 189217 CrossRef
 Title
 Primaldual firstorder methods with \({\mathcal {O}(1/\epsilon)}\) iterationcomplexity for cone programming
 Journal

Mathematical Programming
Volume 126, Issue 1 , pp 129
 Cover Date
 20110101
 DOI
 10.1007/s1010700802616
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Cone programming
 Primaldual firstorder methods
 Smooth optimal method
 Nonsmooth method
 Proxmethod
 Linear programming
 Semidefinite programming
 65K05
 65K10
 90C05
 90C22
 90C25
 Industry Sectors
 Authors

 Guanghui Lan ^{(1)}
 Zhaosong Lu ^{(2)}
 Renato D. C. Monteiro ^{(3)}
 Author Affiliations

 1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 303320205, USA
 2. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
 3. School of ISyE, Georgia Institute of Technology, Atlanta, GA, 30332, USA