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Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points

Author

Listed:
  • E. G. Birgin

    (University of São Paulo)

  • G. Haeser

    (University of São Paulo
    Stanford University)

  • A. Ramos

    (Federal University of Paraná)

Abstract

Augmented Lagrangian methods with convergence to second-order stationary points in which any constraint can be penalized or carried out to the subproblems are considered in this work. The resolution of each subproblem can be done by any numerical algorithm able to return approximate second-order stationary points. The developed global convergence theory is stronger than the ones known for current algorithms with convergence to second-order points in the sense that, besides the flexibility introduced by the general lower-level approach, it includes a loose requirement for the resolution of subproblems. The proposed approach relies on a weak constraint qualification, that allows Lagrange multipliers to be unbounded at the solution. It is also shown that second-order resolution of subproblems increases the chances of finding a feasible point, in the sense that limit points are second-order stationary for the problem of minimizing the squared infeasibility. The applicability of the proposed method is illustrated in numerical examples with ball-constrained subproblems.

Suggested Citation

  • E. G. Birgin & G. Haeser & A. Ramos, 2018. "Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points," Computational Optimization and Applications, Springer, vol. 69(1), pages 51-75, January.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:1:d:10.1007_s10589-017-9937-2
    DOI: 10.1007/s10589-017-9937-2
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    References listed on IDEAS

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    1. Birgin, E. G. & Martinez, J. M. & Ronconi, D. P., 2005. "Optimizing the packing of cylinders into a rectangular container: A nonlinear approach," European Journal of Operational Research, Elsevier, vol. 160(1), pages 19-33, January.
    2. Ernesto G. Birgin & Emerson V. Castelani & André L. M. Martinez & J. M. Martínez, 2011. "Outer Trust-Region Method for Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 142-155, July.
    3. E. Birgin & J. Martínez & L. Prudente, 2015. "Optimality properties of an Augmented Lagrangian method on infeasible problems," Computational Optimization and Applications, Springer, vol. 60(3), pages 609-631, April.
    4. Emerson Castelani & André Martinez & J. Martínez & B. Svaiter, 2010. "Addressing the greediness phenomenon in Nonlinear Programming by means of Proximal Augmented Lagrangians," Computational Optimization and Applications, Springer, vol. 46(2), pages 229-245, June.
    5. E. G. Birgin & L. F. Bueno & J. M. Martínez, 2016. "Sequential equality-constrained optimization for nonlinear programming," Computational Optimization and Applications, Springer, vol. 65(3), pages 699-721, December.
    6. Francisco Facchinei & Stefano Lucidi, 1998. "Convergence to Second Order Stationary Points in Inequality Constrained Optimization," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 746-766, August.
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    Cited by:

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