基于动态差分进化算法的双层多目标优化方法研究

Bi-level multi-objective optimization can more accurately describe hierarchical relationship of real-life problems and more comprehensively embody willingness of decision makers, which commonly appears in many practical problem solving tasks including economics and management, military affairs, transportation, electric power systems, engineering optimization, and others. However, the non-convex nature and non-differentiable in everywhere make the bi-level optimization problems difficult to be solved using numerical methodologies. Especially the multiple solutions of multi-objective optimization problem and complexity of bi-level optimization structure, constructing an effective algorithm for general bi-level multi-objective optimization problems is more challenging. For the nested characteristic of bi-level multi-objective optimization problems, this project intends to design a hierarchical population structure, and then construct a general framework for bi-level evolutionary multi-objective optimization algorithm. Also, in order to improve the computational efficiency, the self-adaptive interacting mechanisms for the evolution population are further studied. Moreover, to improve the convergence performance, the evolution mechanism of original differential evolution (DE) algorithm is modified to update the population dynamically, and the dynamic differential evolution (DDE) algorithm is consequently developed and its convergence proof is also studied based on the random functional analysis method. Thereafter, based on the proposed self-adaptive interacting evolution population structure and dynamic differential evolution algorithm, and combined the Pareto dominated selection mechanism and non-dominated sorting idea, a robust bi-level evolutionary multi-objective optimization algorithm with good convergence performance and low computational complexity is then developed for general bi-level multi-objective optimization problems. The study of this project will not only help expand the application field of evolutionary algorithms, but also have great significance for enrichment and development of bi-level multi-objective optimization theory and methodology and its applications in practice.

双层多目标优化更能描述实际系统的阶层关系和更全面地体现决策者的意愿，在经济、军事、交通、电力和工程等众多领域具有重要的理论意义和应用背景。但双层优化问题的本质非凸性以及非处处可微性，特别是多目标优化解的多样性和双层优化结构的复杂性，使得建立求解一般双层多目标优化问题的有效算法更具挑战性。本项目针对双层多目标优化交互嵌套的特点，拟设计分层递阶进化种群结构，进而建立双层进化多目标优化算法的一般框架，并研究提高算法效率的进化种群自适应交互机制；研究基于动态更新种群策略的高效差分进化算法，并利用随机泛函分析方法证明收敛性；结合自适应交互进化种群结构和动态差分进化算法，设计具有良好收敛性能的双层进化多目标优化算法，着力为一般双层多目标优化问题提供高效和鲁棒的求解方法。这不但有利于拓宽进化算法的应用领域，而且对于丰富和发展双层多目标优化理论与方法及其在实际中的应用具有重要意义。

双层多目标优化更能描述实际系统的阶层关系和更全面地体现决策者的意愿，在经济、军事、交通、电力和工程等众多领域具有重要的理论意义和应用背景。但双层优化问题的本质非凸性以及非处处可微性给其数值求解带来极大的困难，特别是多目标优化解的多样性和双层优化结构的复杂性，建立求解双层多目标优化问题的有效算法更具挑战性。本项目围绕双层多目标优化问题的求解，主要开展了以下研究工作并取得了相应成果：. 1) 针对原创差分进化算法采用静态更新种群方式不利于提高算法效率的不足，提出采用动态种群更新方式的动态差分进化算法，并设计参数自适应策略和改进变异算子以提高算法的鲁棒性和收敛性能，电力系统经济负荷分配应用实例证实了算法的有效性。. 2) 基于Pareto占优概念对选择算子进行改进，将动态差分进化算法拓展到多目标优化，并为保持求得非占优解具有良好的多样性和分布性，提出一种基于拥挤熵的多样性保持策略。Benchmark测试函数和H-two/H-infinity综合控制器多目标优化设计证实了算法的有效性。.3) 针对双层多目标优化交互嵌套的特点，设计了一种分层递阶进化种群结构，进而建立双层进化多目标优化算法的一般框架和模型。基于自适应交互进化种群结构和多目标动态差分进化算法，设计了一种具有良好收敛性能的双层进化多目标优化算法，为一般双层多目标优化问题提供高效和鲁棒的求解方法。. 4) 针对一类上层函数和约束函数不具有凸性和可微性要求而下层函数可微且凸的非线性双层规划问题，通过KKT条件将双层规划问题转换为单层约束非线性规划问题，结合非固定多段映射罚函数法和精确罚函数法对约束条件进行无约束化处理，提出一种改进的动态差分进化算法优化对系列无约束优化问题进行求解。. 5) 针对和声搜索算法逃离局部最优点能力差等不足，基于高斯变异提出了一种求解多目标优化问题的和声搜索算法(GMHS)。该算法对和声存器规则进行改进以提高收敛能力，在音调调节过程中设计了两种不同的带宽以获得更好的全局探索和局部开发能力，引入高斯变异以提高跳出局部最优解的能力和加快收敛速率。. 6) 项目还在基于正态云模型的果蝇优化算法、免疫粒子群优化算法、变尺度伪并行混沌优化算法等智能优化算法以及基于群体智能的群机器人多目标围捕算法方面等进行了一系列探索和研究。