iteration-complexity for cone programming

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, 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. G. Lan (B) School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA e-mail: [email protected] Z. Lu Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada e-mail: [email protected] R. D. C. Monteiro School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA e-mail: [email protected]