Abstract
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.
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References
W.I. Zangwill, “Nonlinear programming via penalty function,” Manangement Science, vol. 13, pp. 334–358, 1967.
S.P. Han and O.L. Mangasrian, “Exact penalty function in nonlinear programming,” Mathematical Programming, vol. 17, pp. 251–269, 1979.
E. Rosenberg, “Globally convergent algorithms for convex programming,” Mathematics of Operational Rresearch, vol. 6, pp. 437–443, 1981.
J.B. Lasserre, “A globally convergent algorithm for exact penalty functions,” European Journal of Opterational Research, vol. 7, pp. 389–395, 1981.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1982.
E. Rosenberg, “Exact penalty functions and stability in locally Lipschitz programming,” Mathematical Programming, vol. 30, pp. 340–356, 1984.
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.
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.
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.
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.
C. Chen and O.L. Mangasarian, “Smoothing methods for convex inequalities and linear complementarity problems,” Mathematical Programming, vol. 71, pp. 51–69, 1995.
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.
S.C. Fang, J.R. Rajasekera, and H.S.J. Tsao, Entropy Optimization and Mathematical Proggramming, Kluwer, 1997.
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.
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.
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Meng, Z., Dang, C. & Yang, X. On the Smoothing of the Square-Root Exact Penalty Function for Inequality Constrained Optimization. Comput Optim Applic 35, 375–398 (2006). https://doi.org/10.1007/s10589-006-8720-6
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DOI: https://doi.org/10.1007/s10589-006-8720-6