混合整数最优控制问题松弛控制解的一种改进取整策略

An Improved Rounding Strategy for Relaxed Control Solutions of Mixed-Integer Optimal Control Problems

摘要: 混合整数最优控制问题(Mixed-Integer Optimal Control Problem, MIOCP)因其包含整数变量而更加复杂，但更能贴近实际应用需求。Sager等人(Math. Program. Vol. 133, 2012)提出了一种松弛–取整策略，将MICOP问题凸松弛为经典最优控制，再对凸松弛的最优解进行取整近似(Sum Up Rounding, SUR)，得到原问题的近似解，并证明了近似误差为时间步长的同阶无穷小。然而，该近似误差估计的同阶无穷小的系数项是时间区间总长度的指数函数，当控制问题的时间区间较大时，这个误差可能会非常大。针对这一问题，本文对SUR策略进行改进，提出一个新的控制取整策略，证明了新控制策略的收敛性，并通过数值例子验证了本文的策略显著提高了MIOCP的求解精度。

Abstract: Mixed-Integer Optimal Control Problems (MIOCP) are more complex due to the inclusion of integer variables but are more aligned with practical application needs. Sager et al. (Math. Program. Vol. 133, 2012) proposed a relaxation-rounding strategy that convexly relaxes the MIOCP to a classical optimal control problem and then approximates the optimal solution of the convex relaxation (Sum Up Rounding, SUR) to obtain an approximate solution to the original problem, proving that the approximation error is of the same order as the infinitesimal of the time step. However, the coefficient of this infinitesimal error estimate is an exponential function of the total length of the time interval, and when the time interval of the control problem is large, this error can be very significant. To address this issue, this paper improves the SUR strategy and proposes a new control rounding strategy, proving the convergence of the new control strategy and verifying through numerical examples that it significantly improves the solution accuracy of MIOCP.