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A homotopy method based on penalty function for nonlinear semidefinite programming

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  • Li Yang
  • Bo Yu
  • YanXi Li

Abstract

This paper proposes a homotopy method based on a penalty function for solving nonlinear semidefinite programming problems. The penalty function is the composite function of an exponential penalty function, the eigenvalue function and a nonlinear operator mapping. Representations of its first and second order derivatives are given. Using the penalty function, a new homotopy is constructed. Global convergence of a smooth curve determined by the homotopy is proven under mild conditions. In the process of numerically tracing the curve, the method requires just the solution of a linear system of dimension $$n+2$$ n + 2 , whereas a homotopy method proposed by Yang and Yu (Comput Optim Appl 56(1):81–96, 2013 ) requires a system of dimension $$n+m(m+1)/2+1$$ n + m ( m + 1 ) / 2 + 1 to be solved, where $$n$$ n is the number of variables while $$m$$ m is the order of constraint matrix. So, it is expected that the proposed method can improve the efficiency of the method proposed by Yang and Yu. Preliminary numerical experiments are presented and show that the considered algorithm is efficient for some nonlinear semidefinite programming problems. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Li Yang & Bo Yu & YanXi Li, 2015. "A homotopy method based on penalty function for nonlinear semidefinite programming," Journal of Global Optimization, Springer, vol. 63(1), pages 61-76, September.
    • Handle: RePEc:spr:jglopt:v:63:y:2015:i:1:p:61-76
    DOI: 10.1007/s10898-015-0276-5
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    References listed on IDEAS

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    1. Huixian Wu & Hezhi Luo & Xiaodong Ding & Guanting Chen, 2013. "Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 56(3), pages 531-558, December.
    2. Hui-juan Xiong & Bo Yu, 2010. "An aggregate deformation homotopy method for min-max-min problems with max-min constraints," Computational Optimization and Applications, Springer, vol. 47(3), pages 501-527, November.
    3. H. Luo & H. Wu & G. Chen, 2012. "On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming," Journal of Global Optimization, Springer, vol. 54(3), pages 599-618, November.
    4. Li Yang & Bo Yu, 2013. "A homotopy method for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 56(1), pages 81-96, September.
    5. Xin Chen & Houduo Qi & Liqun Qi & Kok-Lay Teo, 2004. "Smooth Convex Approximation to the Maximum Eigenvalue Function," Journal of Global Optimization, Springer, vol. 30(2), pages 253-270, November.
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