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Some Investigations on Saddle Points of the Lagrangian Function

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  • Giorgio Giorgi

    (University of Pavia)

Abstract

We take into consideration the classical saddle points conditions of the Lagrangian function for a convex (or concave) minimization problem (maximization problem). After recalling some basic results, we give a survey of several constraint qualifications and regularity conditions assuring that the Kuhn-Tucker saddle points conditions hold. Then we discuss the existence of saddle points conditions for some classes of functions more general than the class of convex (or concave) functions. Finally, we discuss saddle points conditions for programming problems with convex (or concave) constraints, linear affine constraints and a set constraint.

Suggested Citation

  • Giorgio Giorgi, 2024. "Some Investigations on Saddle Points of the Lagrangian Function," DEM Working Papers Series 223, University of Pavia, Department of Economics and Management.
    • Handle: RePEc:pav:demwpp:demwp0223
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    File URL: https://economiaemanagement.dip.unipv.it/sites/dip10/files/2024-10/DEMWP0223.pdf
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    References listed on IDEAS

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    1. D. Gale, 1967. "A Geometric Duality Theorem with Economic Applications," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 34(1), pages 19-24.
    2. Takayama,Akira, 1985. "Mathematical Economics," Cambridge Books, Cambridge University Press, number 9780521314985, January.
    3. X. M. Yang, 2001. "On E-Convex Sets, E-Convex Functions, and E-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 699-704, June.
    4. Alberto Cambini & Laura Martein, 2009. "Generalized Convexity and Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-70876-6, December.
    5. Horst, R., 1984. "On the convexification of nonlinear programming problems: An applications-oriented survey," European Journal of Operational Research, Elsevier, vol. 15(3), pages 382-392, March.
    6. Z. K. Xu, 1997. "Local Saddle Points and Convexification for Nonconvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 739-746, September.
    7. Giorgio Giorgi & Bienvenido Jiménez & Vicente Novo, 2023. "Basic Mathematical Programming Theory," International Series in Operations Research and Management Science, Springer, number 978-3-031-30324-1, April.
    8. J. Gwinner & W. Oettli, 1994. "Theorems of the Alternative and Duality for INF-SUP Problems," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 238-256, February.
    9. E. A. Youness, 1999. "E-Convex Sets, E-Convex Functions, and E-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 439-450, August.
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