Approximate scenario solutions in the progressive hedging algorithm. A numerical study with an application to fisheries management.

*(English)*Zbl 0739.90047This paper focuses on a time-discrete controllable process in time stages \(t=0,1,\dots,T\). The state of the process at time \(t\) is denoted by the variable \(x_ t\). The transition from the state at time \(t\) to that at time \(t+1\) is governed by a control variable \(u_ t\) but is also dependent on an auxiliary variable, the scenario \(s^ t\). The scenarios can be illustrated with a tree structure, the scenario tree. Thus we can describe the transition as follows: \(x_{t+1}=G_ t(x_ t,u_ t,s^ t)\). Probabilities are attached to the scenarios.

To solve a stochastic optimal control problem the so-called ‘progressive hedging algorithm’ developed by R. T. Rockafellar and R. J.-B. Wets [Math. Oper. Res. 16, No. 1, 119-147 (1991; Zbl 0729.90067)] is specialized. The paper describes how the scenario aggregation principle can be combined with approximate solutions of the individual scenario problems, resulting in a computationally efficient algorithm where two individual Lagrangian-based procedures are merged into one.

Computational results are given for an example from fisheries management. Numerical experiments indicate that only crude scenario solutions are needed.

To solve a stochastic optimal control problem the so-called ‘progressive hedging algorithm’ developed by R. T. Rockafellar and R. J.-B. Wets [Math. Oper. Res. 16, No. 1, 119-147 (1991; Zbl 0729.90067)] is specialized. The paper describes how the scenario aggregation principle can be combined with approximate solutions of the individual scenario problems, resulting in a computationally efficient algorithm where two individual Lagrangian-based procedures are merged into one.

Computational results are given for an example from fisheries management. Numerical experiments indicate that only crude scenario solutions are needed.

Reviewer: L.Paditz (Dresden)

##### MSC:

90C15 | Stochastic programming |

91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |

93E20 | Optimal stochastic control |

93C95 | Application models in control theory |

90C90 | Applications of mathematical programming |

90-08 | Computational methods for problems pertaining to operations research and mathematical programming |

93C55 | Discrete-time control/observation systems |

49L20 | Dynamic programming in optimal control and differential games |

##### Keywords:

multistage decision making; time-discrete controllable process; scenario tree; progressive hedging algorithm; scenario aggregation principle; fisheries management
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\textit{T. Helgason} and \textit{S. W. Wallace}, Ann. Oper. Res. 31, 425--444 (1991; Zbl 0739.90047)

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