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Dual Solutions in Convex Stochastic Optimization

Author

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  • Teemu Pennanen

    (Department of Mathematics, King’s College London, London WC2R 2LS, United Kingdom)

  • Ari-Pekka Perkkiö

    (Mathematics Institute, Ludwig Maximilian University of Munich, 80333 Munich, Germany)

Abstract

This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental results in operations research, stochastic optimal control, and financial mathematics. Our analysis builds on the theory of Fréchet spaces of random variables whose topological dual can be identified with the direct sum of another space of random variables and a space of singular functionals. The results are illustrated by deriving sufficient and necessary optimality conditions for several more specific problem classes. We obtain significant extensions to earlier models, for example, on stochastic optimal control, portfolio optimization, and mathematical programming.

Suggested Citation

  • Teemu Pennanen & Ari-Pekka Perkkiö, 2025. "Dual Solutions in Convex Stochastic Optimization," Mathematics of Operations Research, INFORMS, vol. 50(3), pages 2375-2404, August.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:3:p:2375-2404
    DOI: 10.1287/moor.2022.0270
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    References listed on IDEAS

    as
    1. Schachermayer, W., 1992. "A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time," Insurance: Mathematics and Economics, Elsevier, vol. 11(4), pages 249-257, December.
    2. Teemu Pennanen, 2011. "Convex Duality in Stochastic Optimization and Mathematical Finance," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 340-362, May.
    3. Rockafellar, R. T. & Wets, R. J. -B., 1975. "Stochastic convex programming: Kuhn-Tucker conditions," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 349-370, December.
    4. Teemu Pennanen, 2011. "Arbitrage and deflators in illiquid markets," Finance and Stochastics, Springer, vol. 15(1), pages 57-83, January.
    Full references (including those not matched with items on IDEAS)

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