## Approximate scenario solutions in the progressive hedging algorithm. A numerical study with an application to fisheries management.(English)Zbl 0739.90047

This 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.

### 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

Zbl 0729.90067
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### References:

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