一种用于时变优化的简单而高精度的预测-校正算法
A Simple yet Highly Accurate Prediction-Correction Algorithm for Time-Varying Optimization
AI总结：
本文提出一种基于Lagrange插值外推的简单高精度预测-校正算法SHARP，无需Hessian或梯度即可实现O(h^p)跟踪误差，并能跟踪非凸目标函数的平稳点轨迹。
摘要：
本文针对无约束时变优化问题，提出了一种简单而高精度的预测-校正算法SHARP。其预测基于对过去解进行Lagrange插值所导出的外推。由于该外推可以在不需要Hessian矩阵甚至梯度的情况下计算，因此计算成本较低。为了保证预测的稳定性，该算法包含一个接受条件，当更新幅度过大时拒绝该预测。在假设目标轨迹的p阶导数有界且校正步骤的收敛为局部线性的条件下，所提方法实现了O(h^p)的跟踪误差，其中h为采样周期。我们还证明了即使目标函数是非凸的，该方法也能跟踪平稳点的轨迹。数值实验证明了所提算法的高精度。
英文摘要：
This paper proposes a simple yet highly accurate prediction-correction algorithm, SHARP, for unconstrained time-varying optimization problems. Its prediction is based on an extrapolation derived from the Lagrange interpolation of past solutions. Since this extrapolation can be computed without Hessian matrices or even gradients, the computational cost is low. To ensure the stability of the prediction, the algorithm includes an acceptance condition that rejects the prediction when the update is excessively large. The proposed method achieves a tracking error of $O(h^{p})$, where $h$ is the sampling period, assuming that the $p$-th derivative of the target trajectory is bounded and the convergence of the correction step is locally linear. We also prove that the method can track a trajectory of stationary points even if the objective function is non-convex. Numerical experiments demonstrate the high accuracy of the proposed algorithm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
