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Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization

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  • Lv, Jian
  • Pang, Li-Ping
  • Wang, Jin-He

Abstract

We present a proximal bundle method for minimizing the nonconvex maximum eigenvalue function based on a real time control system. The oracle used in our proximal bundle method is able to compute separately the value and subgradient of the outer convex function. Besides, it can also calculate the value and derivatives of the smooth inner mapping. In each iteration, we solve a certain quadratic programming problem in which the smooth inner mapping is replaced by its Taylor-series linearization around the current serious step. By using the backtracking test, we can make a better approximation of the objective function. With no additional assumption, we prove the global convergence of our special bundle method. We present numerical examples demonstrating the efficiency of our algorithm on several feedback control syntheses.

Suggested Citation

  • Lv, Jian & Pang, Li-Ping & Wang, Jin-He, 2015. "Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 635-651.
    • Handle: RePEc:eee:apmaco:v:265:y:2015:i:c:p:635-651
    DOI: 10.1016/j.amc.2015.05.119
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    References listed on IDEAS

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    1. Huiling Lin, 2012. "An inexact spectral bundle method for convex quadratic semidefinite programming," Computational Optimization and Applications, Springer, vol. 53(1), pages 45-89, September.
    2. C. Helmberg & F. Rendl & R. Weismantel, 2000. "A Semidefinite Programming Approach to the Quadratic Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 197-215, June.
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    Cited by:

    1. Fan-Yun Meng & Li-Ping Pang & Jian Lv & Jin-He Wang, 2017. "An approximate bundle method for solving nonsmooth equilibrium problems," Journal of Global Optimization, Springer, vol. 68(3), pages 537-562, July.
    2. 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.
    3. Jian Lv & Li-Ping Pang & Fan-Yun Meng, 2018. "A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information," Journal of Global Optimization, Springer, vol. 70(3), pages 517-549, March.

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