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Solving semi-infinite programs by smoothing projected gradient method


  • Mengwei Xu


  • Soon-Yi Wu


  • Jane Ye



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

    1. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    2. Chen Ling & Qin Ni & Liqun Qi & Soon-Yi Wu, 2010. "A new smoothing Newton-type algorithm for semi-infinite programming," Journal of Global Optimization, Springer, vol. 47(1), pages 133-159, May.
    3. K.L. Teo & X.Q. Yang & L.S. Jennings, 2000. "Computational Discretization Algorithms for Functional Inequality Constrained Optimization," Annals of Operations Research, Springer, vol. 98(1), pages 215-234, December.
    4. Ting-Jang Shiu & Soon-Yi Wu, 2012. "Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems," Computational Optimization and Applications, Springer, vol. 53(1), pages 91-113, September.
    5. 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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    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.


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