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On the Dual Approach to Recursive Optimization

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  • Matthias Messner
  • Nicola Pavoni
  • Sleet Christopher

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 non-additively 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 non-additively 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.

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Bibliographic Info

Paper provided by Carnegie Mellon University, Tepper School of Business in its series GSIA Working Papers with number 2012-E8.

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Date of creation: Oct 2011
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Handle: RePEc:cmu:gsiawp:1772106779

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Postal: Tepper School of Business, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213-3890
Web page: http://www.tepper.cmu.edu/

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Web: http://student-3k.tepper.cmu.edu/gsiadoc/GSIA_WP.asp

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  1. Messner Matthias & Pavoni Nicola & Sleet Christopher, . "On the Dual Approach to Recursive Optimization," GSIA Working Papers 2012-E12, Carnegie Mellon University, Tepper School of Business.
  2. Ramon Marimon & Vincenzo Quadrini, 2006. "Competition, Innovation and Growth with Limited Commitment," NBER Working Papers 12474, National Bureau of Economic Research, Inc.
  3. Mele, Antonio, 2014. "Repeated moral hazard and recursive Lagrangeans," Journal of Economic Dynamics and Control, Elsevier, vol. 42(C), pages 69-85.
  4. 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.
  5. Patrick J. Kehoe & Fabrizio Perri, 2002. "International Business Cycles with Endogenous Incomplete Markets," Econometrica, Econometric Society, vol. 70(3), pages 907-928, May.
  6. 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.
  7. Harold L. Cole & Felix Kubler, 2011. "Recursive Contracts, Lotteries and Weakly Concave Pareto Sets," NBER Working Papers 17064, National Bureau of Economic Research, Inc.
  8. 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.
  9. Yili Chien & Harold Cole & Hanno Lustig, 2011. "A Multiplier Approach to Understanding the Macro Implications of Household Finance," Review of Economic Studies, Oxford University Press, vol. 78(1), pages 199-234.
  10. Matthias Messner & Nicola Pavoni, 2004. "On the Recursive Saddle Point Method," Levine's Bibliography 122247000000000050, UCLA Department of Economics.
  11. Albert Marcet & Thomas J. Sargent & Juha Seppala, 1996. "Optimal taxation without state-contingent debt," Economics Working Papers 170, Department of Economics and Business, Universitat Pompeu Fabra, revised Oct 2001.
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Cited by:
  1. Messner Matthias & Pavoni Nicola & Sleet Christopher, . "On the Dual Approach to Recursive Optimization," GSIA Working Papers 2012-E12, Carnegie Mellon University, Tepper School of Business.
  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. David Martimort & Aggey Semenov & Lars Stole, 2013. "A Theory of Contracts with Limited Enforcement," Working Papers E1304E, University of Ottawa, Department of Economics.
  4. Messner Matthias & Pavoni Nicola & Sleet Christopher, . "Recursive Methods for Dynamic Incentive Problems," GSIA Working Papers 2012-E13, Carnegie Mellon University, Tepper School of Business.

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