AI 中文总结
研究非凸目标与非线性等式约束下的分布式优化难题，提出反馈线性化和奇异摄动框架，构建理想动力学并改进为分布式实现，证明其局部指数收敛，在特定条件下吸引区域可扩展，提升了分布式优化的性能。
AI 中文摘要
当非凸目标与非线性等式约束相结合时，分布式优化变得极具挑战性：可行性是网络耦合的，而大多数现有的指数收敛方法依赖于凸性或仿射约束。本文针对分别涉及分布式局部非线性等式约束和非线性聚合等式约束的两类代表性问题，引入了一种反馈线性化和奇异摄动框架。该框架将共识和可行性残差的调节与沿可行流形的优化分离。具体而言，构建了理想的反馈线性化动力学，其输出行为可明确指定，而其零输出动力学与聚合目标在可行流形上的投影梯度流一致。然而，理想的反馈线性化输入由状态依赖的网络耦合代数方程确定，因此不能以分布式方式直接实现。为克服这一障碍，我们用快速残差跟踪动力学代替非局部代数解，得到一种仅使用局部和相邻信息的奇异摄动实现。在仅对可行流形施加的局部二次增长条件下，我们建立了理想动力学和分布式动力学的局部指数收敛性。对于足够强的时间尺度分离，分布式实现的收敛速率可以选择得任意接近理想动力学的收敛速率。显式欧拉离散化也被证明能保持局部指数收敛。在更强的流形正则性条件下，吸引区域扩展到可行流形的管状邻域，对于仿射局部等式约束，扩展到整个允许状态空间。
英文摘要
Distributed optimization becomes particularly challenging when nonconvex objectives are combined with nonlinear equality constraints: feasibility is network coupled, while most existing exponentially convergent methods rely on convexity or affine constraints. This paper introduces a feedback-linearization and singular-perturbation framework for two representative problem classes involving, respectively, distributed local nonlinear equality constraints and nonlinear aggregate equality constraints. The framework separates the regulation of consensus and feasibility residuals from optimization along the feasible manifold. Specifically, an ideal feedback-linearized dynamics is constructed whose output behavior can be explicitly assigned, while its zero-output dynamics coincides with the projected gradient flow of the aggregate objective on the feasible manifold. The ideal feedback-linearizing input, however, is determined by a state-dependent, network-coupled algebraic equation and is therefore not directly implementable in a distributed manner. To overcome this obstruction, we replace the nonlocal algebraic solution with a fast residual-tracking dynamics, yielding a singular-perturbation realization that uses only local and neighboring information. Under a local quadratic-growth condition imposed only on the feasible manifold, we establish local exponential convergence of both the ideal and distributed dynamics. For sufficiently strong time-scale separation, the convergence rate of the distributed realization can be chosen arbitrarily close to that of the ideal dynamics. Explicit Euler discretizations are also proved to preserve local exponential convergence. Under stronger manifold regularity conditions, the region of attraction extends to a tubular neighborhood of the feasible manifold and, for affine local equality constraints, to the whole admissible state space.
