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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.

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0205, USA

    Guanghui Lan

  2. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Zhaosong Lu

  3. School of ISyE, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Renato D. C. Monteiro

Authors
  1. Guanghui Lan
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  2. Zhaosong Lu
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  3. Renato D. C. Monteiro
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Corresponding author

Correspondence to Guanghui Lan.

Additional information

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.

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Lan, G., Lu, Z. & Monteiro, R.D.C. Primal-dual first-order methods with \({\mathcal {O}(1/\epsilon)}\) iteration-complexity for cone programming. Math. Program. 126, 1–29 (2011). https://doi.org/10.1007/s10107-008-0261-6

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  • DOI: https://doi.org/10.1007/s10107-008-0261-6

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