Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part I: Convex Relaxations
研究控制变量取二值且随时间变化的抛物型偏微分方程最优控制问题，通过分析可行切换模式的凸包构建紧凸松弛，以处理切换次数上限等组合约束，数值示例表明该方法能显著改善直接松弛的对偶界。
.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. While such combinatorial constraints are often seen as an additional complication that is treated in a heuristic postprocessing, the core of our approach is to investigate the convex hull of all feasible switching patterns in order to define a tight convex relaxation of the control problem. The convex relaxation is built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics. A numerical example for the case of a bounded number of switchings shows that our approach can significantly improve the dual bounds given by the straightforward continuous relaxation, which is obtained by relaxing binarity constraints.KeywordsPDE-constrained optimizationswitching time optimizationconvex relaxationsMSC codes90C1190C57