Abstract
In this paper, we introduce a unified framework for the study of penalty concepts by means of the separation functions in the image space (see Ref. 1). Moreover, we establish new results concerning a correspondence between the solutions of the constrained problem and the limit points of the unconstrained minima. Finally, we analyze some known classes of penalty functions and some known classical results about penalization, and we show that they can be derived from our results directly.
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Giannessi, F.,Theorems of Alternative and Optimality Conditions, Journal of Optimization Theory and Applications, Vol. 42, pp. 331–365, 1984.
Giannessi, F.,Metodi Matematici della Programmazione. Problemi Lineari e Non Lineari, Quaderni dell'Unione Matematica Italiana, Pitagora Editrice, Bologna, Italy, 1982.
Auslender, A.,Penalty Methods for Computing Points That Satisfy Second-Order Necessary Conditions, Mathematical Programming, Vol. 17, pp. 229–238, 1979.
Bertsekas, D. P.,Necessary and Sufficient Conditions for a Penalty Method to Be Exact, Mathematical Programming, Vol. 9, pp. 87–99, 1975.
Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, New York, 1968.
Minoux, M.,Mathematical Programming, Wiley, New York, New York, 1987.
Bartholomew-Biggs, M. C.,Recursive Quadratic Programming Methods Based on the Augmented Lagrangian, Mathematical Programming Study, Vol. 31, pp. 21–41, 1987.
Bertsekas, D. P.,Constrained Optimization and Lagrange Multipliers Methods, Academic Press, New York, New York, 1982.
Szego, G. P.,Programmazione Matematica, Minimization Algorithms, Edited by G. P. Szego, Academic Press, New York, New York, 1972.
Fletcher, R.,Penalty Functions, Mathematical Programming Symposium: The State of the Art, Bonn, Germany, 1982; Springer-Verlag, Berlin, Germany, 1982.
Di Pillo, G., andGrippo, L.,A Continuously Differentiable Exact Penalty Function for Nonlinear Programming Problems with Inequality Constraints, SIAM Journal on Control and Optimization, Vol. 23, pp. 72–84, 1985.
Han, S. P., andMangasarian, O. L.,Exact Penalty Functions in Nonlinear Programming, Mathematical Programming, Vol. 17, pp. 251–265, 1979.
Powell, M. J.,Algorithms for Nonlinear Constraints That Use Lagrangian Functions, Mathematical Programming, Vol. 14, pp. 224–248, 1978.
Rockafellar, R. T.,Penalty Methods and Augmented Lagrangians in Nonlinear Programming, Proceedings of the 5th IFIP Conference on Optimization Techniques, Rome, Italy, 1973.
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Communicated by F. Giannessi
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Pappalardo, M. Image space approach to penalty methods. J Optim Theory Appl 64, 141–152 (1990). https://doi.org/10.1007/BF00940028
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DOI: https://doi.org/10.1007/BF00940028