## Hierarchical optimization: An introduction.(English)Zbl 0751.90067

Summary: Decision problems involving multiple agents invariably lead to conflict and gaming. In recent years, multi-agent systems have been analyzed using approaches that explicitly assign to each agent a unique objective function and set of decision variables; the system is defined by a set of common constraints that affects all agents. The decisions made by each agent in these approaches affect the decisions made by the others and their objectives. When strategies are selected simultaneously, in a noncooperative manner, solutions are defined as equilibrium points, so that at optimality no player can do better by unilaterally altering his choice. There are other types of noncooperative decision problems, though, where there is a hierarchical ordering of the agents, and one set has the authority to strongly influence the preferences of the other agents. Such situations are analyzed using a concept known as a Stackelberg strategy. The hierarchical optimization problem conceptually extends the open-loop Stackelberg model to $$K$$ players. In this paper, we provide a brief introduction and survey of recent work in the literature, and summarize the contributions of this volume. It should be noted that the survey is not meant to be exhaustive, but rather to place recent papers in context.

### MSC:

 90C30 Nonlinear programming 90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming 91A65 Hierarchical games (including Stackelberg games) 93A13 Hierarchical systems

### Keywords:

hierarchical optimization; open-loop Stackelberg model
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### References:

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