Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm
研究了控制变量取二值且随时间切换的抛物型偏微分方程最优控制问题，提出一种函数空间中的外逼近算法，证明其迭代在L²中强收敛到凸化问题全局最优解，并用二维数值算例验证效率。
.We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon; they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. In a companion paper [C. Buchheim, A. Grütering, and C. Meyer, SIAM J. Optim., arXiv:2203.07121, 2024], we describe the \(L^p\)-closure of the convex hull of feasible switching patterns as the intersection of convex sets derived from finite-dimensional projections. In this paper, the resulting outer description is used for the construction of an outer approximation algorithm in function space, whose iterates are proven to converge strongly in \(L^2\) to the global minimizer of the convexified optimal control problem. The linear-quadratic subproblems arising in each iteration of the outer approximation algorithm are solved by means of a semismooth Newton method. A numerical example in two spatial dimensions illustrates the efficiency of the overall algorithm.KeywordsPDE-constrained optimizationswitching time optimizationouter approximationMSC codes49M9990C11