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

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

Research output: Working paperDiscussion paperOther research output

291 Downloads (Pure)

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
Place of PublicationTilburg
PublisherEconometrics
Number of pages20
Volume2011-133
Publication statusPublished - 2011

Publication series

NameCentER Discussion Paper
Volume2011-133

Fingerprint

Impulse control
Maximum principle
Optimality conditions
Operations research
Optimal solution
Fixed costs
Sufficiency
Concavity
Optimal control
Cost function
Reviewing
State variable
Jump
Interaction

Keywords

  • Impulse Control Maximum Principle
  • Optimal Control
  • discrete continuous system
  • state-jumps
  • present value formulation.

Cite this

@techreport{a7a2140143e34ecc977a7337fa89b07c,
title = "The Deterministic Impulse Control Maximum Principle in Operations Research: Necessary and Sufficient Optimality Conditions (replaces CentER DP 2011-052)",
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.",
keywords = "Impulse Control Maximum Principle, Optimal Control, discrete continuous system, state-jumps, present value formulation.",
author = "M. Chahim and R.F. Hartl and P.M. Kort",
note = "Subsequently published in the European Journal of Operational Research (2012) Pagination: 20",
year = "2011",
language = "English",
volume = "2011-133",
series = "CentER Discussion Paper",
publisher = "Econometrics",
type = "WorkingPaper",
institution = "Econometrics",

}

The Deterministic Impulse Control Maximum Principle in Operations Research : Necessary and Sufficient Optimality Conditions (replaces CentER DP 2011-052). / Chahim, M.; Hartl, R.F.; Kort, P.M.

Tilburg : Econometrics, 2011. (CentER Discussion Paper; Vol. 2011-133).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - The Deterministic Impulse Control Maximum Principle in Operations Research

T2 - Necessary and Sufficient Optimality Conditions (replaces CentER DP 2011-052)

AU - Chahim, M.

AU - Hartl, R.F.

AU - Kort, P.M.

N1 - Subsequently published in the European Journal of Operational Research (2012) Pagination: 20

PY - 2011

Y1 - 2011

N2 - 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.

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.

KW - Impulse Control Maximum Principle

KW - Optimal Control

KW - discrete continuous system

KW - state-jumps

KW - present value formulation.

M3 - Discussion paper

VL - 2011-133

T3 - CentER Discussion Paper

BT - The Deterministic Impulse Control Maximum Principle in Operations Research

PB - Econometrics

CY - Tilburg

ER -