带罗宾边界椭圆最优控制问题的自适应交替乘子方向方法求解
An Adaptive for Elliptic Optimal Control Problems with Robin Boundary Conditions
摘要: 本文研究带罗宾边界条件的椭圆型最优控制问题。基于有限元离散化方法，将该问题转化为优化问题，并采用自适应乘子交替方向法(adaptive-ADMM)对其进行数值求解。在不依赖拉格朗日乘子存在性的前提下，证明了交替乘子方向方法的收敛性。此外，利用问题对惩罚参数p的敏感性变化，提出了p的自适应调整策略，显著提升了ADMM方法的鲁棒性与收敛速度。本文同时建立了遍历型与非遍历型收敛速率估计，并将所提自适应交替乘子方向方法与采用不同固定值的标准交替乘子方向方法进行对比，验证了该方法的有效性。
Abstract: This paper investigates elliptic optimal control problems with Robin boundary conditions. Based on the finite element discretization method, the problem is transformed into an optimization problem and solved numerically using the adaptive alternating direction method of multipliers (adaptive-ADMM). Without relying on the existence of Lagrange multipliers, the convergence of the ADMM method is proved. Furthermore, by exploiting the sensitivity of the problem to the penalty parameter ρ, an adaptive adjustment strategy for ρ is proposed, which significantly improves the robustness and convergence speed of the ADMM method. Both ergodic and non-ergodic convergence rate estimates are established. The proposed adaptive ADMM method is compared with the standard ADMM method using different fixed parameter values, and the numerical results verify its effectiveness.
文章引用：林星源. 带罗宾边界椭圆最优控制问题的自适应交替乘子方向方法求解[J]. 理论数学, 2026, 16(4): 99-115. https://doi.org/10.12677/PM.2026.164095

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