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On the subdifferential of the value function in economic optimization problems

Author

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  • 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," Post-Print hal-00187221, HAL.
    • Handle: RePEc:hal:journl:hal-00187221
    DOI: 10.1016/0304-4068(95)00717-2
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    References listed on IDEAS

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    1. Mas-Colell, Andreu, 1986. "The Price Equilibrium Existence Problem in Topological Vector Lattice s," Econometrica, Econometric Society, vol. 54(5), pages 1039-1053, September.
    2. Florenzano, Monique, 1983. "On the existence of equilibria in economies with an infinite dimensional commodity space," Journal of Mathematical Economics, Elsevier, vol. 12(3), pages 207-219, December.
    3. Dechert, W. D., 1982. "Lagrange multipliers in infinite horizon discrete time optimal control models," Journal of Mathematical Economics, Elsevier, vol. 9(3), pages 285-302, March.
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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. Marimon, Ramon & Werner, Jan, 2021. "The envelope theorem, Euler and Bellman equations, without differentiability," Journal of Economic Theory, Elsevier, vol. 196(C).
    3. 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.
    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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