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Nonlinear Programming via König’s Maximum Theorem

Author

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  • P. Montiel López

    (University of Granada)

  • M. Ruiz Galán

    (University of Granada)

Abstract

Starting from one extension of the Hahn–Banach theorem, the Mazur–Orlicz theorem, and a not very restrictive concept of convexity, that arises naturally in minimax theory, infsup-convexity, we derive an equivalent version of that fundamental result for finite dimensional spaces, which is a sharp generalization of König’s Maximum theorem. It implies several optimal statements of the Lagrange multipliers, Karush/Kuhn–Tucker, and Fritz John type for nonlinear programs with an objective function subject to both equality and inequality constraints.

Suggested Citation

  • P. Montiel López & M. Ruiz Galán, 2016. "Nonlinear Programming via König’s Maximum Theorem," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 838-852, September.
    • Handle: RePEc:spr:joptap:v:170:y:2016:i:3:d:10.1007_s10957-016-0959-1
    DOI: 10.1007/s10957-016-0959-1
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    References listed on IDEAS

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    1. A. Dax & V. P. Sreedharan, 1997. "Theorems of the Alternative and Duality," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 561-590, September.
    2. T. Illés & G. Kassay, 1999. "Theorems of the Alternative and Optimality Conditions for Convexlike and General Convexlike Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(2), pages 243-257, May.
    3. N. Dinh & V. Jeyakumar, 2014. "Rejoinder on: Farkas’ lemma: three decades of generalizations for mathematical optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 41-44, April.
    4. N. Dinh & V. Jeyakumar, 2014. "Farkas’ lemma: three decades of generalizations for mathematical optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 1-22, April.
    5. V. Jeyakumar & G. M. Lee & G. Li, 2010. "Global Optimality Conditions for Classes of Non-convex Multi-objective Quadratic Optimization Problems," Springer Optimization and Its Applications, in: Regina S. Burachik & Jen-Chih Yao (ed.), Variational Analysis and Generalized Differentiation in Optimization and Control, pages 177-186, Springer.
    6. Jean B. Lasserre, 2009. "Duality and a Farkas lemma for integer programs," Springer Optimization and Its Applications, in: Charles Pearce & Emma Hunt (ed.), Optimization, edition 1, chapter 0, pages 15-39, Springer.
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