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Primal-Dual Nonlinear Rescaling Method for Convex Optimization

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  • R. Polyak

    (George Mason University)

  • I. Griva

    (George Mason University)

Abstract

In this paper, we consider a general primal-dual nonlinear rescaling (PDNR) method for convex optimization with inequality constraints. We prove the global convergence of the PDNR method and estimate the error bounds for the primal and dual sequences. In particular, we prove that, under the standard second-order optimality conditions, the error bounds for the primal and dual sequences converge to zero with linear rate. Moreover, for any given ratio 0 > γ > 1, there is a fixed scaling parameter kγ > 0 such that each PDNR step shrinks the primal-dual error bound by at least a factor 0 > γ > 1, for any k ≥ kγ. The PDNR solver was tested on a variety of NLP problems including the constrained optimization problems (COPS) set. The results obtained show that the PDNR solver is numerically stable and produces results with high accuracy. Moreover, for most of the problems solved, the number of Newton steps is practically independent of the problem size.

Suggested Citation

  • R. Polyak & I. Griva, 2004. "Primal-Dual Nonlinear Rescaling Method for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 111-156, July.
    • Handle: RePEc:spr:joptap:v:122:y:2004:i:1:d:10.1023_b:jota.0000041733.24606.99
    DOI: 10.1023/B:JOTA.0000041733.24606.99
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    References listed on IDEAS

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    1. Roman Polyak, 2001. "Log-Sigmoid Multipliers Method in Constrained Optimization," Annals of Operations Research, Springer, vol. 101(1), pages 427-460, January.
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    Cited by:

    1. Efsun Kürüm & Kasirga Yildirak & Gerhard-Wilhelm Weber, 2012. "A classification problem of credit risk rating investigated and solved by optimisation of the ROC curve," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(3), pages 529-557, September.
    2. Marielena Fonseca Tófoli & Edilaine Martins Soler & Antonio Roberto Balbo & Edméa Cássia Baptista & Leonardo Nepomuceno, 2020. "Interior/exterior-point methods with inertia correction strategy for solving optimal reactive power flow problems with discrete variables," Annals of Operations Research, Springer, vol. 286(1), pages 243-263, March.
    3. Liwei Zhang & Jian Gu & Xiantao Xiao, 2011. "A class of nonlinear Lagrangians for nonconvex second order cone programming," Computational Optimization and Applications, Springer, vol. 49(1), pages 61-99, May.
    4. Da Tian, 2015. "An exterior point polynomial-time algorithm for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 61(1), pages 51-78, May.
    5. Da Tian, 2014. "An entire space polynomial-time algorithm for linear programming," Journal of Global Optimization, Springer, vol. 58(1), pages 109-135, January.
    6. Pinheiro, Ricardo B.N.M. & Lage, Guilherme G. & da Costa, Geraldo R.M., 2019. "A primal-dual integrated nonlinear rescaling approach applied to the optimal reactive dispatch problem," European Journal of Operational Research, Elsevier, vol. 276(3), pages 1137-1153.
    7. M. Gonçalves & J. Melo & L. Prudente, 2015. "Augmented Lagrangian methods for nonlinear programming with possible infeasibility," Journal of Global Optimization, Springer, vol. 63(2), pages 297-318, October.

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