A tutorial on the deterministic impulse control maximum principle: Necessary and sufficient optimality conditions

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

Research output: Contribution to journalArticleScientificpeer-review

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.
Original languageEnglish
Pages (from-to)18-26
JournalEuropean Journal of Operational Research
Volume219
Issue number1
DOIs
Publication statusPublished - 2012

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Impulse Control
Necessary and Sufficient Optimality Conditions
Maximum principle
Maximum Principle
Necessary Optimality Conditions
Sufficiency
Optimal Solution
Necessary Conditions of Optimality
Concave function
Concavity
Costs
Multiple Solutions
Violate
Cost Function
Optimal Control Problem
Jump
Cost functions
Tutorial
Impulse control
Optimality conditions

Cite this

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title = "A tutorial on the deterministic impulse control maximum principle: Necessary and sufficient optimality conditions",
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.",
author = "M. Chahim and R.F. Hartl and P.M. Kort",
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A tutorial on the deterministic impulse control maximum principle : Necessary and sufficient optimality conditions. / Chahim, M.; Hartl, R.F.; Kort, P.M.

In: European Journal of Operational Research, Vol. 219, No. 1, 2012, p. 18-26.

Research output: Contribution to journalArticleScientificpeer-review

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T1 - A tutorial on the deterministic impulse control maximum principle

T2 - Necessary and sufficient optimality conditions

AU - Chahim, M.

AU - Hartl, R.F.

AU - Kort, P.M.

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AB - 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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