Abstract
Under certain sufficient conditions for strict local optimality in a mathematical program, it is well known that a number of non-differentiable penalty functions are locally exact. With sufficient conditions involving the contingent derivative, it is shown that this local exactness is valid for programs whose objective and constraint functions need not be differentiable or even continuous.
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References
Howe, S.,New Conditions for Exactness of a Simple Penalty Function, SIAM Journal on Control, Vol. 11, pp. 378–381, 1973.
Bazaraa, M. S., andGoode, J. J.,Sufficient Conditions for a Globally Exact Penalty Function without Convexity, Mathematical Programmng Study, Vol. 19, pp. 1–15, 1982.
Rosenberg, E.,Exact Penalty Functions and Stability in Locally Lipschitz Programming, Mathematical Programming, Vol. 30, pp. 340–356, 1984.
Aubin, J. P. Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusion and Differential Inclusions, Advances in Mathematics, Supplementary Studies, Vol. 7A, Edited by L. Nachbin, Academic Press, New York, New York, 1981.
Studniarski, M.,Necessary and Sufficient Conditions for Isolated Local Minima of Nonsmooth Functions, SIAM Journal on Control and Optimization, Vol. 24, pp. 1044–1049, 1986.
Ursescu, C.,Tangent Sets' Calculus and Necessary Conditions for Extremality, SIAM Journal on Control and Optimization, Vol. 20, pp. 563–574, 1982.
Michel, P., andPenot, J. P., Calcul Sous-Différentiel pour des Fonctions Lipschitziennes et Non Lipschitziennes, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, France, Vol. 298, Série I, pp. 269–272, 1984.
Penot, J. P.,Variations on the Theme of Nonsmooth Analysis: Another Sub-differential, Nondifferentiable Optimization: Motivations and Applications, Edited by V. F. Demyanov and D. Pallaschke, Springer-Verlag, Berlin, Germany, 1985.
Bazaraa, M. S., Goode, J. J., andNashed, M. Z.,On the Cones of Tangents with Applications to Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 13, pp. 389–426, 1974.
Penot, J. P., Calcul Sous-Différentiel et Optimisation, Journal of Functional Analysis, Vol. 27, pp. 248–276, 1978.
Ward, D. E., andBorwein, J. M.,Nonsmooth Calculus in Finite Dimensions, SIAM Journal on Control and Optimization, Vol. 25, pp. 1312–1340, 1987.
Rockafellar, R. T.,Generalized Directional Derivatives and Subgradients of Nonconvex Functions, Canadian Journal of Mathematics, Vol. 32, pp. 257–280, 1980.
Penot, J. P.,Differentiability of Relations and Differential Stability of Perturbed Optimization Problems, SIAM Journal on Control and Optimization, Vol. 22, pp. 529–551, 1984.
Han, S. P., andMagasarian, O. L.,Exact Penalty Functions in Nonlinear Programming, Mathematical Programming, Vol. 17, pp. 251–269, 1979.
Ward, D. E.,Isotone Tangent Cones and Nonsmooth Optimization, Optimization, Vol. 18, pp. 769–783, 1987.
Zalinescu, C.,On Convex Sets in General Position, Linear Algebra and Applications, Vol. 64, pp. 191–198, 1985.
Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley, New York, New York, 1983.
Balinski, M., andWolfe, P., Editors, Mathematical Programming Study, Vol. 3, p. vi, 1975.
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Communicated by O. L. Mangasarian
The author is grateful to the referees for their helpful comments.
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Ward, D.E. Exact penalties and sufficient conditions for optimality in nonsmooth optimization. J Optim Theory Appl 57, 485–499 (1988). https://doi.org/10.1007/BF02346165
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DOI: https://doi.org/10.1007/BF02346165