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Solving semi-infinite programs by smoothing projected gradient method

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  • Mengwei Xu
  • Soon-Yi Wu
  • Jane Ye

Abstract

In this paper, we study a semi-infinite programming (SIP) problem with a convex set constraint. Using the value function of the lower level problem, we reformulate SIP problem as a nonsmooth optimization problem. Using the theory of nonsmooth Lagrange multiplier rules and Danskin’s theorem, we present constraint qualifications and necessary optimality conditions. We propose a new numerical method for solving the problem. The novelty of our numerical method is to use the integral entropy function to approximate the value function and then solve SIP by the smoothing projected gradient method. Moreover we study the relationships between the approximating problems and the original SIP problem. We derive error bounds between the integral entropy function and the value function, and between locally optimal solutions of the smoothing problem and those for the original problem. Using certain second order sufficient conditions, we derive some estimates for locally optimal solutions of problem. Numerical experiments show that the algorithm is efficient for solving SIP. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:3:p:591-616
    DOI: 10.1007/s10589-014-9654-z
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    References listed on IDEAS

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    6. Dong-Hui Li & Liqun Qi & Judy Tam & Soon-Yi Wu, 2004. "A Smoothing Newton Method for Semi-Infinite Programming," Journal of Global Optimization, Springer, vol. 30(2), pages 169-194, November.
    7. J. J. Ye & S. Y. Wu, 2008. "First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 419-434, May.
    8. 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.
    9. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
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