Constrained Well-Posed Two-Level Optimization Problems

Abstract

A two-level optimization problem corresponding to a two-player game in which player 1 has the leadership in playing the game is considered. Let K ₁ and K ₂ be the sets of admissible strategies for the two players. Player 1 (called the leader) and player 2 (called the follower) must select strategies v ₁ ∈ K ₁ and v ₂ ∈ K ₂ respectively, in order to minimize their objective functionals J ₁ and J ₂. It is supposed that player 1 knows everything about player 2 but player 2 knows only the strategy announced by player 1. So, more precisely, player 1 chooses first an optimal strategy knowing that player 2 will react by playing optimally and that his choice cannot be affected by player 1. This concept, introduced by Von Stackelberg in 1939 [24] in the context of static economic competition has been presented in a control theoretic framework by Chen and Cruz (1972) and Simaan and Cruz (1973). A great deal of papers have been devoted to these problems in static and dynamics games (a good list of references can be found in [3], particularly on dynamic cases, and in [1] for application to economic models).