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On the Dual Approach to Recursive Optimization

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  • Matthias Messner
  • Nicola Pavoni
  • Christopher Sleet

Abstract

We bring together the theories of duality and dynamic programming. We show that the dual of an additively separable dynamic optimization problem can be recursively decomposed using summaries of past Lagrange multipliers as state variables. Analogous to the Bellman decomposition of the primal problem, we prove equality of values and solution sets for recursive and sequential dual problems. In nonadditively separable settings, the equivalence of the recursive and sequential dual is not guaranteed. We relate recursive dual and recursive primal problems. If the Lagrangian associated with a constrained optimization problem admits a saddle then, even in nonadditively separable settings, the values of the recursive dual and recursive primal problems are equal. Additionally, the recursive dual method delivers necessary conditions for a primal optimum. If the problem is strictly concave, the recursive dual method delivers necessary and sufficient conditions for a primal optimum. When a saddle exists, states on the optimal dual path are subdifferentials of the primal value function evaluated at states on the optimal primal path and vice versa.

Suggested Citation

  • Matthias Messner & Nicola Pavoni & Christopher Sleet, 2011. "On the Dual Approach to Recursive Optimization," Working Papers 423, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
  • Handle: RePEc:igi:igierp:423
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    References listed on IDEAS

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    1. Ramon Marimon & Vincenzo Quadrini, 2005. "Competition, innovation and growth with limited commitment," Economics Working Papers 933, Department of Economics and Business, Universitat Pompeu Fabra.
    2. Matthias Messner & Nicola Pavoni & Christopher Sleet, 2012. "Recursive Methods for Incentive Problems," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 15(4), pages 501-525, October.
    3. Kydland, Finn E. & Prescott, Edward C., 1980. "Dynamic optimal taxation, rational expectations and optimal control," Journal of Economic Dynamics and Control, Elsevier, vol. 2(1), pages 79-91, May.
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    5. S. Rao Aiyagari & Albert Marcet & Thomas J. Sargent & Juha Seppala, 2002. "Optimal Taxation without State-Contingent Debt," Journal of Political Economy, University of Chicago Press, vol. 110(6), pages 1220-1254, December.
    6. Harold Cole & Felix Kubler, 2012. "Recursive Contracts, Lotteries and Weakly Concave Pareto Sets," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 15(4), pages 479-500, October.
    7. Patrick J. Kehoe & Fabrizio Perri, 2002. "International Business Cycles with Endogenous Incomplete Markets," Econometrica, Econometric Society, vol. 70(3), pages 907-928, May.
    8. Matthias Messner & Nicola Pavoni, 2016. "On the Recursive Saddle Point Method," Dynamic Games and Applications, Springer, vol. 6(2), pages 161-173, June.
    9. Daron Acemoglu & Mikhail Golosov & Aleh Tsyvinski, 2010. "Dynamic Mirrlees Taxation under Political Economy Constraints," Review of Economic Studies, Oxford University Press, vol. 77(3), pages 841-881.
    10. Mele, Antonio, 2014. "Repeated moral hazard and recursive Lagrangeans," Journal of Economic Dynamics and Control, Elsevier, vol. 42(C), pages 69-85.
    11. Messner Matthias & Pavoni Nicola & Sleet Christopher, "undated". "On the Dual Approach to Recursive Optimization," GSIA Working Papers 2012-E12, Carnegie Mellon University, Tepper School of Business.
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    Cited by:

    1. Matthias Messner & Nicola Pavoni & Christopher Sleet, 2012. "Recursive Methods for Incentive Problems," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 15(4), pages 501-525, October.
    2. Messner Matthias & Pavoni Nicola & Sleet Christopher, "undated". "Recursive Methods for Dynamic Incentive Problems," GSIA Working Papers 2012-E13, Carnegie Mellon University, Tepper School of Business.
    3. Nicola Pavoni & Christopher Sleet & Matthias Messner, 2014. "The Dual Approach to Recursive Optimization: Theory and Examples," 2014 Meeting Papers 1267, Society for Economic Dynamics.
    4. Messner Matthias & Pavoni Nicola & Sleet Christopher, "undated". "On the Dual Approach to Recursive Optimization," GSIA Working Papers 2012-E12, Carnegie Mellon University, Tepper School of Business.
    5. David Martimort & Aggey Semenov & Lars Stole, 2017. "A Theory of Contracts with Limited Enforcement," Review of Economic Studies, Oxford University Press, vol. 84(2), pages 816-852.

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