, Volume 126, Issue 1, pp 129
First online:
Primaldual firstorder methods with \({\mathcal {O}(1/\epsilon)}\) iterationcomplexity 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 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.
Keywords
Cone programming Primaldual firstorder methods Smooth optimal method Nonsmooth method Proxmethod Linear programming Semidefinite programmingMathematics Subject Classification (2000)
65K05 65K10 90C05 90C22 90C25 Title
 Primaldual firstorder methods with \({\mathcal {O}(1/\epsilon)}\) iterationcomplexity for cone programming
 Journal

Mathematical Programming
Volume 126, Issue 1 , pp 129
 Cover Date
 201101
 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