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Optimality Functions and Lopsided Convergence

Author

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  • Johannes O. Royset

    (Naval Postgraduate School)

  • Roger J-B Wets

    (University of California, Davis)

Abstract

Optimality functions pioneered by E. Polak characterize stationary points, quantify the degree with which a point fails to be stationary, and play central roles in algorithm development. For optimization problems requiring approximations, optimality functions can be used to ensure consistency in approximations, with the consequence that optimal and stationary points of the approximate problems indeed are approximately optimal and stationary for an original problem. In this paper, we review the framework and illustrate its application to nonlinear programming and other areas. Moreover, we introduce lopsided convergence of bifunctions on metric spaces and show that this notion of convergence is instrumental in establishing consistency of approximations. Lopsided convergence also leads to further characterizations of stationary points under perturbations and approximations.

Suggested Citation

  • Johannes O. Royset & Roger J-B Wets, 2016. "Optimality Functions and Lopsided Convergence," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 965-983, June.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:3:d:10.1007_s10957-015-0839-0
    DOI: 10.1007/s10957-015-0839-0
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    References listed on IDEAS

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    1. Johannes Royset, 2013. "On sample size control in sample average approximations for solving smooth stochastic programs," Computational Optimization and Applications, Springer, vol. 55(2), pages 265-309, June.
    2. Raghu Pasupathy, 2010. "On Choosing Parameters in Retrospective-Approximation Algorithms for Stochastic Root Finding and Simulation Optimization," Operations Research, INFORMS, vol. 58(4-part-1), pages 889-901, August.
    3. J. O. Royset & E. Y. Pee, 2012. "Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 855-882, December.
    4. Johannes O. Royset & Roberto Szechtman, 2013. "Optimal Budget Allocation for Sample Average Approximation," Operations Research, INFORMS, vol. 61(3), pages 762-776, June.
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