Abstract
Exploiting the image-space approach, we give an overview of regularity conditions. A notion of regularity for the image of a constrained extremum problem is given. The relationship between image regularity and other concepts is also discussed. It turns out that image regularity is among the weakest conditions for the existence of normal Lagrange multipliers.
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Giannessi, F.,Theorems of the Alternative and Optimality Conditions, Journal of Optimization Theory and Applications, Vol. 42, pp. 331–365, 1984.
Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, John Wiley, New York, New York, 1966.
Dien, P. H., Mastroeni, G., Pappalardo, M., andQuang, P. H.,Regularity Conditions for Constrained Extremum Problems via Image Space: the Linear Case (to appear).
Martein, L.,Regularity Conditions for Constrained Extremum Problems, Journal of Optimization Theory and Applications, Vol. 47, pp. 217–233, 1985.
Dien, P. H., andSach, P. H.,Further Properties of the Regularity of Inclusion Systems, Nonlinear Analysis: Theory, Methods and Applications, Vol. 3, pp. 1251–1267, 1989.
Dien, P. H.,On the Regularity Condition for the Extremal Problem under Locally Lipschitz Inclusion Constraints, Applied Mathematics and Optimization, Vol. 13, pp. 151–161, 1985.
Mangasarian, O. L., andFromovitz, S.,The Fritz-John Necessary Optimality Condition in the Presence of Equality and Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 7, pp. 37–47, 1967.
Kurcyusz, S.,On the Existence and Nonexistence of Lagrange Multipliers in a Banach Space, Journal of Optimization Theory and Applications, Vol. 20, pp. 81–110, 1976.
Zowe, J., andKurcyusz, S.,Regularity and Stability for the Mathematical Programming Problem in Banach Spaces, Applied Mathematics and Optimization, Vol. 5, pp. 49–62, 1979.
Robinson, S. M.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.
Penot, J. P.,A New Constraint Qualification Condition, Journal of Optimization Theory and Applications, Vol. 48, pp. 459–468, 1986.
Clarke, F.,Optimization and Nonsmooth Analysis, J. Wiley, New York, New York, 1984.
Cambini, A.,Nonlinear Separation Theorem, Duality, and Optimality Condition, Optimization and Related Fields, Edited by R. Conti, E. De Giorgi, and F. Giannessi, Springer-Verlag, Berlin, Germany, pp. 57–93, 1984.
Quang, P. H.,Lagrange Multiplier Rules via Image Space Analysis, Nonsmooth Optimization: Theory and Methods, Edited by F. Giannessi, Gordon and Breach, London, England, pp. 354–365, 1993.
Tardella, F.,On the Image of a Constrained Extremum Problem and Some Applications to the Existence of a Minimum, Journal of Optimization Theory and Applications, Vol. 60, pp. 93–104, 1989.
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Dien, P.H., Mastroeni, G., Pappalardo, M. et al. Regularity conditions for constrained extremum problems via image space. J Optim Theory Appl 80, 19–37 (1994). https://doi.org/10.1007/BF02196591
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DOI: https://doi.org/10.1007/BF02196591