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Constrained well-posed two-level optimization problems. (English) Zbl 0786.90112
Nonsmooth optimization and related topics, Proc. 4th Course Int. Sch. Math., Erice/Italy 1988, Ettore Majorana Int. Sci. Ser., Phys. Sci. 43, 307-325 (1989).
[For the entire collection see Zbl 0719.00020.]
This article is aimed at characterizing two-level optimization problems of Stackelberg type, stated as follows: let $$K_ 1$$ and $$K_ 2$$ be the sets of admissible strategies of the two players, where player 1 (the leader) and player 2 (the follower) must select strategies $$v_ 1\subset K_ 1$$ and $$v_ 2\subset K_ 2$$, so as to minimize their objective functionals $$J_ 1$$ and $$J_ 2$$. $$K_ i$$ is assumed to be a nonempty subset of a topological space $$V_ i$$, and the functional $$J_ i$$ is defined on $$V_ 1\times V_ 2$$ and valued in $${\mathbf R}$$. The leader chooses first an optimal strategy knowing that the follower will react optimally.
Let $$M_ 2(v_ 1)= \{\bar v_ 2\subset K_ 2; J_ 2(v_ 1,\bar v_ 2)=\inf_{v_ 2\in K_ 2} J_ 2(v_ 1,v_ 2)\}$$ be the reaction set of the follower which is reduced to a singleton in case of uniqueness of the optimization problem of player 2. In case of multiplicity, the leader has to protect himself against the worst choice of the follower; then the leader faces of the problem $$(\text{S}):\text{Min}_{v_ 1\subset K_ 1}\text{Sup}_{v_ 2\in M_ 2(v_ 1)} J_ 1(v_ 1,v_ 2)$$. The existence of a solution of problem (S) needs a compactness hypothesis on spaces $$V_ i$$. The convergence towards a solution can be obtained therefore with respect to the weak topology. The author wants to characterize the class of problems ensuring strong convergence. Hence the notion of well-posed two-level optimization problems is used.
A characterization of these problems is given in terms of minimizing sequences whether or not the reaction set is a singleton. An application to the constrained linear quadratic dynamic games is presented.

##### MSC:
 91A65 Hierarchical games (including Stackelberg games) 90C90 Applications of mathematical programming 91A23 Differential games (aspects of game theory) 91A05 2-person games