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.

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.