Optimality conditions and algorithms for parameter design problems with two-level structure

We consider the parameter design problem for a central system coordinating plural semiautonomous subsystems each of which optimizes its own objective under the given parameter from the center. The center makes a decision of the parameter values to be assigned to the subsystems so as to optimize its objective, considering the values of optimized subsystems' performances. Such a parameter design problem is formulated in the framework of a two-level planning problem and becomes an optimization problem including optimal-value functions, and accordingly, a nondifferentiable optimization problem. In this paper, based on Gauvin's studies concerned with the directional derivatives of optimal-value functions, we derive the necessary conditions for the parameter design problem by means of a new theorem of the alternative. The results obtained here are slightly different from the Kuhn-Tucker-like conditions, and are adapted to the structure of the problem. As the computational method for our problem, we propose applying an existing generalized gradient algorithm called the "bundle method" in a class of nondifferentiable optimization methods, and also show a numerical example.