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A regularization method for constrained nonlinear least squares

Author

Listed:
  • Dominique Orban

    (École Polytechnique)

  • Abel Soares Siqueira

    (Federal University of Paraná)

Abstract

We propose a regularization method for nonlinear least-squares problems with equality constraints. Our approach is modeled after those of Arreckx and Orban (SIAM J Optim 28(2):1613–1639, 2018. https://doi.org/10.1137/16M1088570) and Dehghani et al. (INFOR Inf Syst Oper Res, 2019. https://doi.org/10.1080/03155986.2018.1559428) and applies a selective regularization scheme that may be viewed as a reformulation of an augmented Lagrangian. Our formulation avoids the occurrence of the operator $$A(x)^T A(x)$$A(x)TA(x), where A is the Jacobian of the nonlinear residual, which typically contributes to the density and ill conditioning of subproblems. Under boundedness of the derivatives, we establish global convergence to a KKT point or a stationary point of an infeasibility measure. If second derivatives are Lipschitz continuous and a second-order sufficient condition is satisfied, we establish superlinear convergence without requiring a constraint qualification to hold. The convergence rate is determined by a Dennis–Moré-type condition. We describe our implementation in the Julia language, which supports multiple floating-point systems. We illustrate a simple progressive scheme to obtain solutions in quadruple precision. Because our approach is similar to applying an SQP method with an exact merit function on a related problem, we show that our implementation compares favorably to IPOPT in IEEE double precision.

Suggested Citation

  • Dominique Orban & Abel Soares Siqueira, 2020. "A regularization method for constrained nonlinear least squares," Computational Optimization and Applications, Springer, vol. 76(3), pages 961-989, July.
    • Handle: RePEc:spr:coopap:v:76:y:2020:i:3:d:10.1007_s10589-020-00201-2
    DOI: 10.1007/s10589-020-00201-2
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    References listed on IDEAS

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    1. M. Fernanda & P. Costa & Edite Fernandes, 2005. "A primal-dual interior-point algorithm for nonlinear least squares constrained problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(1), pages 145-166, June.
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    3. Narges Bidabadi & Nezam Mahdavi-Amiri, 2014. "Superlinearly Convergent Exact Penalty Methods with Projected Structured Secant Updates for Constrained Nonlinear Least Squares," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 154-190, July.
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    7. A. Izmailov & M. Solodov & E. Uskov, 2015. "Combining stabilized SQP with the augmented Lagrangian algorithm," Computational Optimization and Applications, Springer, vol. 62(2), pages 405-429, November.
    8. Z.F. Li & M.R. Osborne & T. Prvan, 2002. "Adaptive Algorithm for Constrained Least-Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 423-441, August.
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    Cited by:

    1. Ernesto G. Birgin, 2020. "Preface of the special issue dedicated to the XII Brazilian workshop on continuous optimization," Computational Optimization and Applications, Springer, vol. 76(3), pages 615-619, July.

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