Solving Stackelberg equilibrium for multi objective aerodynamic shape optimization

Abstract

• The Stackelberg game is introduced into multi-objective aerodynamic shape optimization . • Numerical methods coupled (EAs) and/or adjoint method with a Stackelberg game are Presented. • Virtual Nash and virtual Stackelberg problems are equivalent, and their solutions are the same. • Final equilibrium solutions are only related to what information is exchanged. • The equilibrium of the weak form Stackelberg problem is as the same as the Nash equilibrium. Numerical methods coupling evolutionary algorithms and/or adjoint methods with a two level hierarchical game between leader and follower, called the Stackelberg game, are presented for solving multi objective aerodynamic shape optimization problems in this paper, in which both leader and follower can be a Nash sub-game with multiple players. The decision making process of leader and follower is hierarchically and non-cooperatively in numerical implementation, more accurately they are alternate. The leader enounces a decision (a Nash sub-game solution), and the follower takes into account this decision and solves the other Nash sub-game optimization problem. The elitist information that follower passes to leader is the response function of follower to leader’s decision, instead of its response only. Then, the leader optimizes his objective based on elitist information accepted from follower depending on his behavior. The approximate response function is obtained by recalculating the follower’s Nash sub-game with a small perturbation on leader’s decision. Finally, a multi-objective shape optimization problem of hypersonic air-breathing vehicle is solved by using the optimization algorithm presented in this paper, and the converged result of the algorithm is proved to be a Stackelberg equilibrium.