Computational Optimization and Applications

, Volume 35, Issue 3, pp 375–398 | Cite as

On the Smoothing of the Square-Root Exact Penalty Function for Inequality Constrained Optimization



In this paper we propose two methods for smoothing a nonsmooth square-root exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem and of the original optimization problem. We develop an algorithm for solving the optimization problem based on the smoothed penalty function and prove the convergence of the algorithm. The efficiency of the smoothed penalty function is illustrated with some numerical examples, which show that the algorithm seems efficient.


constrained optimization penalty function exact penalty function smoothing method ∈-feasible solution, optimal solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W.I. Zangwill, “Nonlinear programming via penalty function,” Manangement Science, vol. 13, pp. 334–358, 1967.Google Scholar
  2. 2.
    S.P. Han and O.L. Mangasrian, “Exact penalty function in nonlinear programming,” Mathematical Programming, vol. 17, pp. 251–269, 1979.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Rosenberg, “Globally convergent algorithms for convex programming,” Mathematics of Operational Rresearch, vol. 6, pp. 437–443, 1981.MATHCrossRefGoogle Scholar
  4. 4.
    J.B. Lasserre, “A globally convergent algorithm for exact penalty functions,” European Journal of Opterational Research, vol. 7, pp. 389–395, 1981.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1982.Google Scholar
  6. 6.
    E. Rosenberg, “Exact penalty functions and stability in locally Lipschitz programming,” Mathematical Programming, vol. 30, pp. 340–356, 1984.MATHMathSciNetGoogle Scholar
  7. 7.
    G. Di Pillo and L. Grippo, “An exact penalty function method with global conergence properties for nonlinear programming problems,” Mathemathical Programming, vol. 36, pp. 1–18, 1986.MathSciNetGoogle Scholar
  8. 8.
    G. Di Pillo and L. Grippo, “On the exactness of a class of nondifferentiable penalty function,” Journal of Optimization Theory and Applications, vol. 57, pp. 385–406, 1988.MathSciNetCrossRefGoogle Scholar
  9. 9.
    S.A. Zenios, M.C. Pinar, and R.S. Dembo, “A smooth penalty function algorithm for network-structured problems,” European Journal of Operational Research, vol. 64, pp. 258–277, 1993.CrossRefGoogle Scholar
  10. 10.
    M.C. Pinar and S.A. Zenios, “On smoothing exact penalty functions for convex constarained optimization,” SIAM Journal on Optimization, vol. 4, pp. 486–511, 1994.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Chen and O.L. Mangasarian, “Smoothing methods for convex inequalities and linear complementarity problems,” Mathematical Programming, vol. 71, pp. 51–69, 1995.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    X.Q. Yang, Z.Q. Meng, X.X. Huang, and G.T.Y. Pong, “Smoothing nonlinear penalty functions for constrained optimization,” Numerical Functional Analysis and Optimization, vol. 24, pp. 351–364, 2003.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    S.C. Fang, J.R. Rajasekera, and H.S.J. Tsao, Entropy Optimization and Mathematical Proggramming, Kluwer, 1997.Google Scholar
  14. 14.
    L. Qi, S.Y. Wu, and G. Zhou, “Semismooth newton methods for solving semi-infinite programming problems,” Journal of Global Optimization, vol. 27, pp. 215–232, 2003.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    K.L. Teo, X.Q. Yang, and L.S. Jennings, “Computational discretization algorithms for functional inequality constrained optimization,” Annals of Operations Research, vol. 98, pp. 215–234, 2000.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.College of Business and AdministrationZhejiang University of TechnologyHangzhouChina
  2. 2.Department of Manufacturing Engineering & Engineering ManagementCity University of Hong KongKowloonHong Kong
  3. 3.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung HomHong Kong

Personalised recommendations