Constraints handling in Nash/Adjoint optimization methods for multi-objective aerodynamic design

Game theory and its particular Nash games are gaining importance in multi-objective optimization in engineering problems over the past decade. Among ongoing advances in optimization methods and tools, many applications including mathematical modeling with constraints in different areas remain challenges in industrial design environments. This paper describes a constraint handling algorithm to extend the use of Nash games to a more realistic multi-objective aerodynamic optimization when taking into account constraints. A competitive Nash game with constraints is adapted to a cooperative game, in which Nash procedure acts as a sub-game to calculate the equilibrium point and collaborates with a player in charge of constraints. The partial gradient of each objective instead of design variables is used as elitist information exchanged symmetrically in sub-game. Existence and equivalence of the solution are analyzed and proved based on Brouwer fixed point theorem. Numerical experiments on 2D multi-objective aerodynamic optimization with lift and/or geometry constraints are implemented, and results show that constraints are satisfied.