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On the Dual Approach to Recursive Optimization

  • Matthias Messner
  • Nicola Pavoni
  • Christopher Sleet

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.

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Paper provided by IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University in its series Working Papers with number 423.

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Date of creation: 2011
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Handle: RePEc:igi:igierp:423
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  1. 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.
  2. 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.
  3. 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.
  4. YiLi Chien & Harold Cole & Hanno Lustig, 2007. "A Multiplier Approach to Understanding the Macro Implications of Household Finance," NBER Working Papers 13555, National Bureau of Economic Research, Inc.
  5. Antonio Mele, 2008. "Repeated Moral Hazard and Recursive Lagrangeans," 2008 Meeting Papers 482, Society for Economic Dynamics.
  6. 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.
  7. Patrick J. Kehoe & Fabrizio Perri, 2000. "International business cycles with endogenous incomplete markets," Staff Report 265, Federal Reserve Bank of Minneapolis.
  8. 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.
  9. Matthias Messner & Nicola Pavoni, 2004. "On the Recursive Saddle Point Method," Levine's Bibliography 122247000000000050, UCLA Department of Economics.
  10. Ramon Marimon & Vincenzo Quadrini, 2006. "Competition, Innovation and Growth with Limited Commitment," NBER Working Papers 12474, National Bureau of Economic Research, Inc.
  11. 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.
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