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A Trajectory-Based Method for Constrained Nonlinear Optimization Problems

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  • M. Montaz Ali

    (University of the Witwatersrand (Wits)
    University of the Witwatersrand (Wits))

  • Terry-Leigh Oliphant

    (University of the Witwatersrand (Wits))

Abstract

A trajectory-based method for solving constrained nonlinear optimization problems is proposed. The method is an extension of a trajectory-based method for unconstrained optimization. The optimization problem is transformed into a system of second-order differential equations with the aid of the augmented Lagrangian. Several novel contributions are made, including a new penalty parameter updating strategy, an adaptive step size routine for numerical integration and a scaling mechanism. A new criterion is suggested for the adjustment of the penalty parameter. Global convergence properties of the method are established.

Suggested Citation

  • M. Montaz Ali & Terry-Leigh Oliphant, 2018. "A Trajectory-Based Method for Constrained Nonlinear Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 177(2), pages 479-497, May.
    • Handle: RePEc:spr:joptap:v:177:y:2018:i:2:d:10.1007_s10957-018-1274-9
    DOI: 10.1007/s10957-018-1274-9
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    References listed on IDEAS

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    1. Jean-Philippe Vial & Israel Zang, 1977. "Unconstrained Optimization by Approximation of the Gradient Path," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 253-265, August.
    2. F. Alvarez & A. Cabot, 2004. "Steepest Descent with Curvature Dynamical System," Journal of Optimization Theory and Applications, Springer, vol. 120(2), pages 247-273, February.
    3. Ernesto Birgin & J. Martínez, 2012. "Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization," Computational Optimization and Applications, Springer, vol. 51(3), pages 941-965, April.
    4. VIAL, Jean-Philippe & ZANG, Israel, 1977. "Unconstrained optimization by approximation of the gradient path," LIDAM Reprints CORE 329, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. S.-P. Han, 1986. "On the Augmented Lagrangian," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 161-168, February.
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