Abstract

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Author information

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Jean-Philippe Chancelier

Université Paris-Est, CERMICS - ENPC, 77455 Marne la Vallée 2, France

Keywords

  • Fenchel-Moreau conjugacy
  • coupling
  • envelope function
  • generalized inf-convolution
  • Bellman equation

Mathematics Subject Classification

46N10, 47N10

References

  1. J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Modern Birkhäuser Classics, Birkhäuser, Boston (2009).https://doi.org/10.1007/978-0-8176-4848-0Google Scholar
  2. H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edition, CMS Books in Mathematics, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-48311-5Google Scholar
  3. D. P. Bertsekas, S. E. Shreve: Stochastic Optimal Control: The Discrete-Time Case, Athena Scientific, Belmont (1996).Google Scholar
  4. A. Cabot, A. Jourani, L. Thibault: Envelopes for sets and functions II: generalized polarity and conjugacy, preprint (2018).https://doi.org/10.1112/S0025579316000309Google Scholar
  5. P. Carpentier, J.-P. Chancelier, G. Cohen, M. De Lara: Stochastic Multi-Stage Optimization. At the Crossroads between Discrete Time Stochastic Control and Stochastic Programming, Springer, Berlin (2015).https://doi.org/10.1007/978-3-319-18138-7Google Scholar
  6. P. Girardeau, V. Leclère, A. B. Philpott: On the convergence of decomposition methods for multistage stochastic convex programs, Math. Oper. Res. 40(1) (2014) 130–145.https://doi.org/10.1287/moor.2014.0664Google Scholar
  7. V. L. Levin: Abstract convexity and the Monge-Kantorovich duality, in: Generalized Convexity and Related Topics, Lecture Notes Econ. Math. Systems 583, Springer, Berlin (2007) 33–72.https://doi.org/10.1007/978-3-540-37007-9_2Google Scholar
  8. M. Loève: Probability Theory I, 4th edition, Springer, New York (1977).https://doi.org/10.1007/978-1-4684-9464-8Google Scholar
  9. J. E. Martínez-Legaz: Generalized convex duality and its economic applications, in: Handbook of Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas, S. Komlósi, S. S. Schaible (eds.), Nonconvex Optimization and Its Applications 76, Springer (2005) 237–292.https://doi.org/10.1007/0-387-23393-8_6Google Scholar
  10. M.-L. Mazure, M. Volle: Équations inf-convolutives et conjugaison de Moreau-Fenchel, Ann. Fac. Sc. Toulouse 12(1) (1991) 103–125.https://doi.org/10.5802/afst.721Google Scholar
  11. J. J. Moreau: Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl. (9), 49 (1970) 109–154.Google Scholar
  12. D. Pallaschke, S. Rolewicz: Foundations of Mathematical Optimization, Mathematics and its Applications 388, Kluwer, Dordrecht (1997).https://doi.org/10.1007/978-94-017-1588-1Google Scholar
  13. T. Pennanen, A.-P. Perkkiö: Shadow price of information in discrete time stochastic optimization, Math. Programming 168(1) (2018) 347–367.https://doi.org/10.1007/s10107-017-1163-2Google Scholar
  14. R. Rockafellar: Integrals which are convex functionals, Pacific J. Math. 24(3) (1968) 525–539.https://doi.org/10.2140/pjm.1968.24.525Google Scholar
  15. R. T. Rockafellar: Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966) 81–89.https://doi.org/10.1215/S0012-7094-66-03312-6Google Scholar
  16. R. T. Rockafellar: Integrals which are convex functionals II, Pacific J. Math. 39(2) (1971) 439–469.https://doi.org/10.2140/pjm.1971.39.439Google Scholar
  17. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  18. R. T. Rockafellar, P. R. Wolenski: Envelope representations in Hamilton-Jacobi theory for fully convex problems of control, in: Proc. 40th IEEE Conference on Decision and Control, vol. 3 (2001) 2768–2771.https://doi.org/10.1109/CDC.2001.980692Google Scholar
  19. A. Rubinov: Abstract Convexity and Global Optimization, Nonconvex Optimization and its Applications 44, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9Google Scholar
  20. F. Santambrogio: Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications 87, Birkhäuser, Basel (2015).https://doi.org/10.1007/978-3-319-20828-2Google Scholar
  21. A. Shapiro: Analysis of stochastic dual dynamic programming method, European J. Oper. Res. 209(1) (2011) 63–72.https://doi.org/10.1016/j.ejor.2010.08.007Google Scholar
  22. I. Singer: Abstract Convex Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York (1997).Google Scholar
  23. M. Volle: Multiapplications duales et problèmes d’optimisation en dualité, C. R. Acad. Sci. Paris, Sér. I, 296 (1983) 11–14.Google Scholar

Suggested citation

J.-P. Chancelier, M. De Lara. “Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to the Stochastic Bellman Equation.” Journal of Convex Analysis 26 (2019), No. 3, 945–966. https://doi.org/10.68381/jca26051

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.