多智能体对等网络中的分布式优化问题指多个智能体根据两两之间能否通信组成对等网络，每个智能体（节点）拥有局部目标函数，整个网络的全局目标函数为所有节点局部目标函数之和。节点需要利用局部目标函数并通过与邻居不断通信来优化全局目标函数。该问题在无线传感器网络的分布式参数估计、分布式资源配置、多智能体强化学习、大规模机器学习等领域有着广泛应用，具有实际应用价值和理论研究价值。本文针对该问题的主要研究成果和创新点如下：(1) 针对分布式算法中节点间通信花费时间长、成本高的问题，提出了一种基于新息压缩的分布式算法COLD，其中新息为待发送变量与其估计值的差异，具有幅值小且收敛至0等优点，解决了直接压缩无法精确收敛的问题。对于光滑强凸目标函数，COLD能够取得线性收敛速度，且适用于有偏压缩器。进一步结合动态缩放技术提出了Dyna-COLD，可支持单比特量化器且保持线性收敛速度。(2) 针对分布式同步算法中节点互相等待导致的闲置时间长、鲁棒性差等问题，提出了两种异步算法AsySPA和APPG，其中节点无需相互等待，可根据自身节奏更新，提高了系统效率。对于非光滑凸函数，AsySPA通过设计自适应步长解决异步模式带来的信息延迟和错位更新问题，并通过引入虚拟节点和增广图证明了算法的精确收敛性以及次线性收敛速度。对于光滑强凸目标函数，APPG通过梯度跟踪方法来估计全局目标函数梯度，从而实现了精确线性收敛速度。(3) 针对已有算法无法实现全局超线性收敛的难点，提出了分布式自适应牛顿法DAN和DAN-LA。DAN引入了分布式有限时间一致性算法以突破已有渐近一致性算法只能取得线性收敛速度的瓶颈，并结合自适应步长以避免高成本的分布式线搜索。DAN-LA进一步设计了基于低秩矩阵近似的方法压缩待传输Hessian矩阵，将节点间通信复杂度降低到与一阶算法一致的同时保持全局超线性收敛速度。(4) 针对特殊场景对分布式算法的要求设计了不同算法。在水下等环境中节点无法通过GPS等方式获取精确位置信息，因此提出了一种基于相对状态符号信息的分布式算法，并通过惩罚函数法证明了收敛性。在分布式资源分配问题中，节点决策变量具有全局约束，因此通过对偶理论将其转化为无约束分布式优化问题并结合有向图上的算法进行求解，基于LMI分析给出了收敛性保证。 以上理论结果均通过数值仿真实验进行了验证并对比已有工作展示了优势。

The distributed optimization problem over multi-agent peer-to-peer networks refers to that multiple agents form a peer-to-peer communication network according to whether they can communicate with each other. Each agent (node) has a local objective function, and the global objective function of the network is the sum of all local objective functions. This problem finds many applications in, e.g.,distributed estimation of wireless sensor networks, distributed resource allocation, multi-agent reinforcement learning, and large-scale machine learning, and has importance in both theory and practice. The main results and contributions of this thesis are summarized as follows. (1) We propose a distributed algorithm COLD based on innovation compression to tackle the problem of slow speed and high cost of communications among nodes in distributed algorithms, where the innovation is defined as the difference between a variable to be sent and its estimate. The innovation has the advantages of small amplitude and convergence to 0, and the compression of innovations solves the problem that direct compression cannot achieve exact convergence. For smooth and strongly convex objective functions, COLD has linear convergence rate, and supports biased compressors. Furthermore, we propose Dyna-COLD by combining COLD with the dynamic scaling method, which supports one-bit binary quantizer and maintains a linear convergence rate.(2) To tackle the problem of long idle time and weak robustness in distributed synchronous algorithms where nodes have to wait for each other, we propose two asynchronous algorithms AsySPA and APPG, where nodes need not wait for any other node and can update at their own pace, and thus the system efficiency is improved. For non-smooth convex objective functions, AsySPA designs an adaptive stepsize to solve the problem of information staleness and out-of-order updates in asynchronous algorithms. We introduce virtual nodes and augmented graphs to show that AsySPA has exact convergence with sublinear convergence rate. For Lipschitz smooth and strongly convex objective functions, APPG estimates the gradient of global objective functions via gradient tracking methods, and achieves exact linear convergence rates.(3) We propose two distributed adaptive Newton methods DAN and DAN-LA with superlinear convergence rates, which have not been achieved in the literature. DAN is obtained by firstly designing a distributed finite-time consensus algorithm to break the bottleneck of linear convergence rate of existing asymptotic consensus algorithms, and integrating it with an adaptive stepsize to avoid expensive distributed line search methods. DAN-LA further combines a Hessian compression method based on matrix low-rank approximation, which reduces the communication cost to that of first order methods, and maintains a globally superlinear convergence rate.(4) We propose algorithms for special applications. For environments such as underwater, forest, and tunnel where accurate positions cannot be obtained through GPS and other methods, we propose a distributed algorithm based on the sign of relative state information. We show its convergence via penalty methods. In the distributed resource allocation problem, nodes have global constraints, and we transform it into an unconstrained distributed optimization problem by the duality theory. Then, we solve it by a distributed optimization algorithm over directed networks, and show its convergence by LMI analysis.