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On the subdifferential of the value function in economic optimization problems

Author

Listed:
  • Jean-Marc Bonnisseau

    (CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Cuong Le Van

    (RFEM - Recherche fondamentale en économie mathématique - CEPREMAP - Centre pour la recherche économique et ses applications - ECO ENS-PSL - Département d'économie de l'ENS-PSL - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

Abstract

The purpose of this paper is to provide a unified treatment to find sufficient conditions for the existence of a subgradient of the value function associated with a convex optimization problem. We recall basic results in convex programming with linear constraints. In particular, the subdifferential of the value function is the opposite of the set of multipliers associated with a solution. We state two results on the non-emptiness of the subdifferential of the value function. The first one is known and the second one is original since we do not assume any continuity condition on the objective function. We apply these results to different cases arising in mathematical economics. The last part is devoted to the case with equality and inequality constraints. We provide a necessary and sufficient condition for the non-emptiness of the subdifferential of the value function which works even if the interior of the positive cone is empty.

Suggested Citation

  • Jean-Marc Bonnisseau & Cuong Le Van, 1996. "On the subdifferential of the value function in economic optimization problems," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00187221, HAL.
    • Handle: RePEc:hal:cesptp:hal-00187221
    DOI: 10.1016/0304-4068(95)00717-2
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    Cited by:

    1. Cuong Van & Raouf Boucekkine & Cagri Saglam, 2007. "Optimal Control in Infinite Horizon Problems: A Sobolev Space Approach," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 32(3), pages 497-509, September.
    2. Olivier Morand & Kevin Reffett & Suchismita Tarafdar, 2018. "Generalized Envelope Theorems: Applications to Dynamic Programming," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 650-687, March.
    3. Marimon, Ramon & Werner, Jan, 2021. "The envelope theorem, Euler and Bellman equations, without differentiability," Journal of Economic Theory, Elsevier, vol. 196(C).
    4. Jean-Michel Grandmont, 2013. "Tribute to Cuong Le Van," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 5-10, March.
    5. Morand, Olivier & Reffett, Kevin & Tarafdar, Suchismita, 2015. "A nonsmooth approach to envelope theorems," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 157-165.

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