基于变分分析的多智能体分布式优化算法研究

Distributed multi-agent optimization is widely used in industrial manufacturing, artificial intelligence and many other scientific and engineering applications, and is an important research direction in the field of automation. In the distributed algorithm, it is the variational property of the mapping near its fixed point that determines convergence and convergence rate of the variable being in some neighborhood of the limit point after many iterations. The latter reflects the general change tendency of the distributed algorithm, which is an important part of the theoretical analysis and has guiding significance for the algorithm design. In general, the whole mapping in a distributed algorithm is composed of gradients of agents’ objective functions and the weight matrix of the network. Since the variational properties of such mappings are beyond the scope of convex analysis, it is necessary to introduce variational analysis theory to discuss. In this project, variational analysis is introduced into the research of distributed optimization. For consensus-based smooth convex optimization and generalized optimal smooth resource allocation, distributed convergent algorithms with fixed stepsizes will be designed, which will also be generalized to distributed nonsmooth convex optimization. Moreover, theory and method in variational analysis including metric subregularity, Lipschitz continuity of solution mappings, stability of constrained systems, error bound of variational inequalities, regularization techniques will be used to obtain the convergence rate and the conditions for exponential convergence. Also, quantitative characterization for asymptotic properties and precise convergence mechanism of distributed algorithms with fixed stepsizes will be obtained.

多智能体分布式优化广泛应用于工业制造、人工智能等诸多科学与工程实际中，是自动化领域重要的研究方向。分布式算法中映射在不动点附近的变分性质，决定了变量经过多次迭代到达极限点某个邻域后的收敛性、收敛速度等，而后者反映出分布式算法的总体变化趋势，是理论分析的重要组成部分，也对算法设计具有指导意义。分布式算法中的映射往往包含个体目标函数梯度的罗列和网络权重矩阵，其变分性质已经超出凸分析范畴，有必要借助变分分析理论进行论述。为此，本项目将变分分析引入到分布式优化研究中，针对分布式一致性光滑凸优化问题和广义资源分配光滑凸优化问题，设计定步长且收敛的分布式求解算法，并推广到分布式非光滑凸优化。利用变分分析理论与方法中的度量次正则性、解映射Lipschitz连续性、约束系统稳定性、变分不等式误差界、正则化技术等，给出算法确切的收敛速度和指数收敛条件，得到分布式定步长算法渐近性质的定量刻画与精确收敛机理。

多智能体分布式优化与博弈具有广泛的应用场景，是自动化领域重要的研究方向。本项目主要研究和取得的成果如下。1) 提出采用变分分析的理论与方法对分布式优化与博弈算法中的复杂映射进行分析，给出指数收敛判据与设计和分析方法；2)把握算法精度与收敛速度的相互制约关系，以牺牲求解精度为代价，提出基于奇异摄动理论的分布式次优快速收敛算法；3)考虑智能体受信道容量和数据存储、计算能力的限制，开展了针对量化信息下分布式优化与均衡算法研究。通过本项目的研究，得到基于变分分析的先进分布式算法设计与理论分析方法，特别是这些算法的指数收敛分析，解决了一类分布式一致性光滑凸优化问题、广义资源分配光滑凸优化问题和博弈均衡计算问题，建立了一套基于变分分析的先进分布式算法设计方案和理论分析方法。相关成果发表在控制领域顶刊IEEE Transactions on Automatic Control、Automatica以及其他国内外期刊和会议上。项目的研究成果为分布式优化与博弈的进一步发展做出了贡献，能够进一步推动在工业生产、人工智能等领域的实际应用。