关于非线性最优控制中重叠Schwarz分解的收敛性
On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control
AI总结：
本文研究了用于求解非线性最优控制问题的重叠Schwarz分解算法的收敛性，证明了其局部线性收敛性，并展示了重叠区域大小对收敛速率的指数影响，同时建立了二次规划的全局收敛性结果。
摘要：
我们研究了用于求解非线性最优控制问题（OCPs）的重叠Schwarz分解算法的收敛性质。该算法将时间域分解为一组重叠子域，并并行求解所有子域上的子问题。通过在重叠子域边界更新对偶信息来实现收敛。我们证明该算法具有局部线性收敛性，且重叠区域大小增加时收敛速率呈指数提升。同时，我们为一般二次规划建立了全局收敛性结果，这使得Schwarz方案可以应用于二次优化算法（如序列二次规划）。我们的收敛性分析的理论基础是非线性OCPs的灵敏度结果，称为“灵敏度指数衰减”（EDS）。直观上，EDS表明在域边界（即初始和终端时间）上的扰动对解的影响随着进入域的深入而呈指数衰减。在此，我们扩展了文献中已有的分析，证明在均匀二次充分条件、可控性条件和有界条件下的非线性OCPs的对偶解也满足EDS。我们通过四旋翼运动规划问题和PDE控制问题的实验验证了我们的理论，并展示了该方法比ADMM更高效，与集中求解器Ipopt同样高效。
英文摘要：
We study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The algorithm decomposes the time domain into a set of overlapping subdomains, and solves all subproblems defined over subdomains in parallel. The convergence is attained by updating primal-dual information at the boundaries of overlapping subdomains. We show that the algorithm exhibits local linear convergence, and that the convergence rate improves exponentially with the overlap size. We also establish global convergence results for a general quadratic programming, which enables the application of the Schwarz scheme inside second-order optimization algorithms (e.g., sequential quadratic programming). The theoretical foundation of our convergence analysis is a sensitivity result of nonlinear OCPs, which we call "exponential decay of sensitivity" (EDS). Intuitively, EDS states that the impact of perturbations at domain boundaries (i.e. initial and terminal time) on the solution decays exponentially as one moves into the domain. Here, we expand a previous analysis available in the literature by showing that EDS holds for both primal and dual solutions of nonlinear OCPs, under uniform second-order sufficient condition, controllability condition, and boundedness condition. We conduct experiments with a quadrotor motion planning problem and a PDE control problem to validate our theory; and show that the approach is significantly more efficient than ADMM and as efficient as the centralized solver Ipopt.
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