, Volume 126, Issue 1, pp 1-29
Date: 06 Feb 2009

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

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Abstract

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.

The work of G. Lan and R. D. C. Monteiro was partially supported by NSF Grants CCF-0430644 and CCF-0808863 and ONR Grants N00014-05-1-0183 and N00014-08-1-0033. Z. Lu was supported in part by SFU President’s Research Grant and NSERC Discovery Grant.