TY - JOUR
T1 - Primal-dual first-order methods with O(1/∈) iteration-complexity for cone programming
AU - Lan, Guanghui
AU - Lu, Zhaosong
AU - Monteiro, Renato D C
PY - 2011/1/1
Y1 - 2011/1/1
N2 - 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.
AB - 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.
KW - Cone programming
KW - Linear programming
KW - Nonsmooth method
KW - Primal-dual first-order methods
KW - Prox-method
KW - Semidefinite programming
KW - Smooth optimal method
UR - http://www.scopus.com/inward/record.url?scp=78651417720&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=78651417720&partnerID=8YFLogxK
U2 - 10.1007/s10107-008-0261-6
DO - 10.1007/s10107-008-0261-6
M3 - Article
AN - SCOPUS:78651417720
SN - 0025-5610
VL - 126
SP - 1
EP - 29
JO - Mathematical Programming
JF - Mathematical Programming
IS - 1
ER -