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Stability of Indices in the KKT Conditions and Metric Regularity in Convex Semi-Infinite Optimization

Author

Listed:
  • M. J. Cánovas

    (Miguel Hernández University of Elche)

  • A. Hantoute

    (Miguel Hernández University of Elche)

  • M. A. López

    (University of Alicante)

  • J. Parra

    (Miguel Hernández University of Elche)

Abstract

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context, Cánovas et al. (SIAM J. Optim. 18:717–732, [2007]) introduced a sufficient condition (called ENC in the present paper) for the strong Lipschitz stability of the optimal set mapping. Now, we show that ENC also entails high stability for the minimal subsets of indices involved in the KKT conditions, yielding a nice behavior not only for the optimal set mapping, but also for its inverse. Roughly speaking, points near optimal solutions are optimal for proximal parameters. In particular, this fact leads us to a remarkable simplification of a certain expression for the (metric) regularity modulus given in Cánovas et al. (J. Glob. Optim. 41:1–13, [2008]) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, [2000]; Control Cybern. 32:543–554, [2003])), which provides a key step in further research oriented to find more computable expressions of this regularity modulus.

Suggested Citation

  • M. J. Cánovas & A. Hantoute & M. A. López & J. Parra, 2008. "Stability of Indices in the KKT Conditions and Metric Regularity in Convex Semi-Infinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(3), pages 485-500, December.
    • Handle: RePEc:spr:joptap:v:139:y:2008:i:3:d:10.1007_s10957-008-9407-1
    DOI: 10.1007/s10957-008-9407-1
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    Cited by:

    1. Li-Ping Pang & Jian Lv & Jin-He Wang, 2016. "Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 433-465, June.
    2. Mengwei Xu & Soon-Yi Wu & Jane Ye, 2014. "Solving semi-infinite programs by smoothing projected gradient method," Computational Optimization and Applications, Springer, vol. 59(3), pages 591-616, December.
    3. M. Goberna & M. Todorov & V. Vera de Serio, 2012. "On stable uniqueness in linear semi-infinite optimization," Journal of Global Optimization, Springer, vol. 53(2), pages 347-361, June.
    4. N. Huy & D. Kim, 2013. "Lipschitz behavior of solutions to nonconvex semi-infinite vector optimization problems," Journal of Global Optimization, Springer, vol. 56(2), pages 431-448, June.
    5. N. Q. Huy & J.-C. Yao, 2011. "Semi-Infinite Optimization under Convex Function Perturbations: Lipschitz Stability," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 237-256, February.
    6. Alexander Y. Kruger & Marco A. López, 2012. "Stationarity and Regularity of Infinite Collections of Sets. Applications to Infinitely Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 390-416, November.

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