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

• 成果类型: Article
• 署名作者: Lan, Guanghui; Lu, Zhaosong; Monteiro, Renato D. C.
• 署名单位: University System of Georgia; Georgia Institute of Technology; Simon Fraser University; University System of Georgia; Georgia Institute of Technology
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-008-0261-6
• 发表日期: 2011
• 页码: 1-29

摘要

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.