SIAM Journal on Control and Optimization, Volume 58, Issue 3, Page 1371-1402, January 2020.
We extend recent work on solving mixed-integer nonlinear optimal control problems (MIOCPs) to the case of integer control functions subject to constraints that involve a pointwise coupling of the state with the integer controls. We extend a theorem due to [S. Sager, H. Bock, and M. Diehl, Math. Program. Ser. A, 133 (2012), pp. 1--23] to the case of MIOCPs with constraints on the integer control and show that the integrality gap vanishes in function space when the coarseness of the rounding grid is driven to zero even after adding constraints of this type. For the time-discretized problem, we extend a sum-up rounding (SUR) scheme due to [S. Sager, C. Reinelt, and H. Bock, Math. Program. Ser. A, 118 (2009), pp. 109--149] to the new problem class. Our scheme permits one to constructively obtain an $\varepsilon$-feasible and $\varepsilon$-optimal binary feasible control. We derive new, tight upper bounds on the integer control approximation error made by SUR. For unconstrained binary controls on equidistant grids, we reduce the approximation error bound from $\mathcal{O}(|\Omega|)$ to $\mathcal{O}(\log |\Omega|)$ asymptotically for $|\Omega| \to \infty$ and a fixed coarseness of the rounding grid, where $|\Omega|$ is the number of binary controls. For constrained binary controls, we show that the approximation problem is more difficult, and we give a proof of an approximation error bound of complexity $\mathcal{O}(|\Omega|)$. A numerical example compares our approach to a state-of-the-art mixed-integer nonlinear programming solver and illustrates the applicability of our results when solving MIOCPs using the direct and simultaneous approach.

中文翻译：

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约束混合整数最优控制的逼近性质和紧边界

SIAM控制与优化杂志，第58卷，第3期，第1371-1402页，2020年1月。
我们将解决混合整数非线性最优控制问题（MIOCP）的最新工作扩展到受约束的整数控制函数的情况，约束涉及状态与整数控件的点状耦合。由于[S。Sager，H。Bock和M.Diehl，数学。程序。老师 A，133（2012），第1--23页]对具有整数控制约束的MIOCP的情况，表明即使舍入网格后，将舍入网格的粗糙度驱动为零时，函数空间的完整性缺口也消失了这种类型的。对于时间离散问题，由于[S。Sager，C。Reinelt和H.Bock，数学。程序。老师 A，118（2009），109--149页]。我们的方案允许一个人以建设性的方式获得一个可行的变量和最优的二元可行控制。我们推导了SUR产生的整数控制近似误差的新的严格上限。对于等距网格上的无约束二进制控制，我们将逼近$ | \ Omega的近似误差从$ \ mathcal {O}（| \ Omega |）$减小为$ \ mathcal {O}（\ log | \ Omega |）$ | \ to \ infty $和舍入网格的固定粗糙度，其中$ | \ Omega | $是二进制控件的数量。对于受约束的二进制控件，我们证明了逼近问题更加困难，并且给出了复杂度$ \ mathcal {O}（| \ Omega |）$的逼近误差界的证明。