Multi-objective Optimization Strategies Using Adjoint Method and Game Theory in Aerodynamics

Abstract

There are currently three different game strategies originated in economics: (1) Cooperative games (Pareto front), (2) Competitive games (Nash game) and (3) Hierarchical games (Stackelberg game). Each game achieves different equilibria with different performance, and their players play different roles in the games. Here, we introduced game concept into aerodynamic design, and combined it with adjoint method to solve multicriteria aerodynamic optimization problems. The performance distinction of the equilibria of these three game strategies was investigated by numerical experiments. We computed Pareto front, Nash and Stackelberg equilibria of the same optimization problem with two conflicting and hierarchical targets under different parameterizations by using the deterministic optimization method. The numerical results show clearly that all the equilibria solutions are inferior to the Pareto front. Non-dominated Pareto front solutions are obtained, however the CPU cost to capture a set of solutions makes the Pareto front an expensive tool to the designer.

Basic Concepts of Game Theory for Aerospace Engineering Applications

“…The numerical results and comparisons with other game theoretical protocols can be found in Tang [10] and Tang et al [11]: they show clearly that all the noncooperative equilibria solutions, namely Nash and Stackelberg equilibria, are inferior to the Pareto front, which is also stable with respect to different parametrizations. Nevertheless the Central Processing Unit (CPU) cost to capture a set of solutions makes the Pareto front an expensive optimizer to the designer, whereas Nash and Stackelberg equilibria are still good satisfactory solutions of the multiobjective optimization problem and are efficient to achieve.…”

“…In the paper by Tang [10] three different solutions originated in economics for treating multiobjective optimization problems are considered: the Pareto solution concept (scalarization approach) in the context of cooperative games, the Nash equilibrium concept in the context of noncooperative games, and the Stackelberg solution concept in the context of hierarchical games. In the following we illustrate the second type of resolution based on the Nash equilibrium concept.…”

Distributed optimization using virtual and real game strategies for multi-criterion aerodynamic design

“…Here, we follow the main idea developed in refs. [20,21] and extend it into more practical multi-objective aerodynamic designs and provide detailed numerical implementation. Several kinds of airfoil splitting and design cases are shown for the virtual and real game strategies.…”

“…[20,21], we introduced a novel approach combining adjoint-variable technique with a formulation derived from game theory [5] to treat multi-point airfoil optimization problems. In a symmetric Nash game, each player attempts to optimize one's own target with exchange of symmetric information with the others [5] .…”

“…, each player optimizes his own criterion by modifying his own subset (design variables) while keeping the other subsets unchanged with exchange of information with others[4,20,21] .Let us consider N players optimizing a set of N objectives ( ) distributed among the players in such a way that each player handles a subset of the set of optimization variables. Let ( )…”