Abstract
We propose a scheme to solve constrained optimization problems by combining a nonlinear penalty method and a descent method. A sequence of nonlinear penalty optimization problems is solved to generate a sequence of stationary points, i.e., each point satisfies a first-order necessary optimality condition of a nonlinear penalty problem. Under some conditions, we show that any limit point of the sequence satisfies the first-order necessary condition of the original constrained optimization problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
RUBINOV, A. M., GLOVER, B. M., and YANG, X.Q., Modified Lagrange and Penalty Functions in Continuous Optimization, Optimization, Vol. 46, pp. 327-351, 1999.
RUBINOV, A. M., YANG, X. Q., and GLOVER, B. M., Nonlinear Unconstrained Optimization Methods: A Review, Progress in Optimization: Contributions from Australasia, Edited by X. Q. Yang et al., Kluwer Academic Publishers, Dordrecht, Holland, pp. 65-77, 2000.
RUBINOV, A. M., GLOVER, B. M., and YANG, X.Q., Decreasing Functions with Applications to Penalization, SIAM Journal on Optimization, Vol. 10, pp. 211-230, 1999.
YANG, X.Q., An Exterior-Point Method for Computing Points that Satisfy Second-Order Necessary Conditions for a C1,1 Optimization Problem, Journal of Mathematical Analysis and Applications, Vol. 87, pp. 118-133, 1994.
CHARALAMBOUS, C., On Conditions for Optimality of the Nonlinear l1-Problem, Mathematical Programming, Vol. 17, pp. 123-135, 1979.
AUSLENDER, A., Penalty Methods for Computing Points that Satisfy Second-Order Necessary Conditions, Mathematical Programming, Vol. 17, pp. 229-238, 1979.
FLETCHER, R., and WATSON, G.A., First and Second-Order Conditions for a Class of Nondifferentiable Optimization Problems, Mathematical Programming, Vol. 18, pp. 291-307, 1980.
WOMERSLEY, R. S., Optimality Conditions for Piecewise Smooth Functions, Mathematical Programming Study, Vol. 17, pp. 13-27, 1982.
BEN-TAL, A. and ZOWE, J., Necessary and Sufficient Optimality Conditions for a Class of Nonsmooth Minimization Problems, Mathematical Programming, Vol. 24, pp. 70-91, 1982.
JEYAKUMAR, V., Composite Nonsmooth Programming with Gateaux Differentiability, SIAM Journal on Optimization, Vol. 1, pp. 30-41, 1991.
JEYAKUMAR, V., and YANG, X. Q., Convex Composite Multiobjective Nonsmooth Programming, Mathematical Programming, Vol. 59, pp. 325-343, 1993.
YANG, X. Q., and JEYAKUMAR, V., First and Second-Order Optimality Conditions for Convex Composite Multiobjective Optimization, Journal of Optimization Theory and Applications, Vol. 91, pp. 209-224, 1997.
YANG, X. Q., Second-Order Global Optimality Conditions for Convex Composite Optimization, Mathematical Programming, Vol. 81, pp. 327-347, 1998.
BORWEIN, J. M., and PREISS, D., A Smooth Variational Principle with Applications to Subdifferentiability and Differentiability, Transactions of the American Mathematical Society, Vol. 303, pp. 517-527, 1987.
CLARKE, F. H., LEAYAEV, YU. S., and WOLENSKI, P.R., Proximal Analysis and Minimization Principles, Journal of Mathematical Analysis and Applications, Vol. 196, 722-735, 1995.
AUBIN, J. P., and FRANKOWSKA, H., Set-Valued Analysis, Birkhauser, Boston, Massachusetts, 1990.
CLARKE, F.H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, NY, 1983.
ROCKAFELLAR, R. T., and WETS, R.J.B., Variational Analysis, Springer Verlag, Berlin, Germany, 1998.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Huang, X., Yang, X. Convergence Analysis of a Class of Nonlinear Penalization Methods for Constrained Optimization via First-Order Necessary Optimality Conditions. Journal of Optimization Theory and Applications 116, 311–332 (2003). https://doi.org/10.1023/A:1022503820909
Issue Date:
DOI: https://doi.org/10.1023/A:1022503820909