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The Deterministic Impulse Control Maximum Principle in Operations Research: Necessary and Sufficient Optimality Conditions (replaces CentER DP 2011-052)

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  • Chahim, M.
  • Hartl, R.F.
  • Kort, P.M.

    (Tilburg University, Center for Economic Research)

Abstract

This paper considers a class of optimal control problems that allows jumps in the state variable. We present the necessary optimality conditions of the Impulse Control Maximum Principle based on the current value formulation. By reviewing the existing impulse control models in the literature, we point out that meaningful problems do not satisfy the sufficiency conditions. In particular, such problems either have a concave cost function, contain a fixed cost, or have a control-state interaction, which have in common that they each violate the concavity hypotheses used in the sufficiency theorem. The implication is that the corresponding problem in principle has multiple solutions that satisfy the necessary optimality conditions. Moreover, we argue that problems with fixed cost do not satisfy the conditions under which the necessary optimality conditions can be applied. However, we design a transformation, which ensures that the application of the Impulse Control Maximum Principle still provides the optimal solution. Finally, we show for the first time that for some existing models in the literature no optimal solution exists.

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Bibliographic Info

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2011-133.

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Date of creation: 2011
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Handle: RePEc:dgr:kubcen:2011133

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Web page: http://center.uvt.nl

Related research

Keywords: Impulse Control Maximum Principle; Optimal Control; discrete continuous system; state-jumps; present value formulation.;

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References

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  1. Luhmer, Alfred, 1986. "A continuous time, deterministic, nonstationary model of economic ordering," European Journal of Operational Research, Elsevier, vol. 24(1), pages 123-135, January.
  2. Gaimon, Cheryl, 1986. "An impulsive control approach to deriving the optimal dynamic mix of manual and automatic output," European Journal of Operational Research, Elsevier, vol. 24(3), pages 360-368, March.
  3. Chahim, M. & Brekelmans, R.C.M. & Hertog, D. den & Kort, P.M., 2011. "An Impulse Control Approach to Dike Height Optimization (Replaced by CentER DP 2012-079)," Discussion Paper 2011-097, Tilburg University, Center for Economic Research.
  4. Kort, Peter, 1988. "Optimal Dynamic Investment Policies of a Value Maximizing Firm," Open Access publications from Tilburg University urn:nbn:nl:ui:12-154529, Tilburg University.
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