混合动态对策理论及大规模约束优化方法研究

Nash equilibrium theory was more and more used in multi-criteria optimization in engineering problems over the past decade. Although many advances have been realized, most applications do not address constrained problem. This project attempts to provide constraint handling method to extend traditional Nash strategy to more realistic multi-criteria optimizations taking into account constraints. A competitive Nash game with constraints is transformed into a cooperative game, in which Nash strategy is a sub-game to calculate the equilibrium, and collaborates with players in charge of constraints. The partial gradient of each objective computed by gradient based methods (i.e. adjoint method) instead of design variables, is used as elitist information exchange in sub-game. Then the famous Brouwer fixed point theorem will be introduced to prove the existence and convergence of the equilibrium of the hybrid game, as well as the equivalence of the hybrid game solution with that of original constrained Nash competitive game. Finally, this new constrained Nash optimization algorithms will be used in stability-constrained aerodynamic shape optimization to verify the validation and reliability of algorithm in dealing with large scale constraints in engineering optimization problems.

自从申请者2000年将经济学中的NASH(纳什)均衡理论发展应用于气动优化设计中以来，对策论优化算法已被广泛应用于各种工程优化问题中，然而NASH对策中对称性的信息交换会改变由每个局中人所单独满足的约束数值。本项目旨在发展大规模约束对策论优化算法，即将有约束的NASH竞争对策转化成无约束的混合对策，原NASH竞争策略及所有约束条件成了混合对策的局中人，而原来的NASH竞争对策成了混合对策的一个子对策。并试图应用布劳威尔不动点定理证明我们发展的新算法解的存在性和收敛性、以及与原约束NASH竞争对策解的等价性。进一步将该约束算法发展应用于由稳定性约束的飞行器气动优化设计问题中，以检验该算法在解决实际复杂工程问题中的可靠性和可行性。

在本项目中我们发展了大规模约束对策论优化算法，即将有约束的NASH竞争对策转化成无约束的混合对策，原NASH竞争策略及所有约束条件成了混合对策的局中人，而原来的NASH竞争对策成了混合对策的一个子对策。并应用布劳威尔不动点定理和NASH均衡理论证明了我们发展的新算法解的存在性和收敛性、以及与原约束NASH竞争对策解的等价性。并将该约束算法应用于由各种约束条件（复杂几何约束、积分形式的气动约束、组合约束等）组成的复杂多约束飞行器多目标气动优化设计问题中，检验了该算法在解决实际复杂工程问题中的可靠性和可行性。