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

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 first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method [9, 11], Nesterov’s smooth approximation scheme [11], and Nemirovski’s prox-method [8], 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 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 (SDP) 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 [2, 3] for solving a set of randomly generated SDP instances.