We consider the problem of designing a cloak for waves described by the Helmholtz equation from an integer programming point of view. The problem can be modeled as a PDE-constrained optimization problem with integer-valued control inputs that are distributed in the computational domain. A first-discretize-then-optimize approach results in a large-scale mixed-integer nonlinear program that is in general intractable because of the large number of integer variables that arise from the discretization of the domain. Instead, we propose an efficient algorithm that is able to approximate the local infima of the underlying nonconvex infinite-dimensional problem arbitrarily close without the need to solve the discretized finite-dimensional integer programs to optimality. We optimize only the continuous relaxations of the approximations for local minima and then apply the sum-up rounding methodology to obtain integer-valued controls. If the solutions of the discretized continuous relaxations converge to a local minimizer of the continuous relaxation, then the resulting discrete-valued control sequence converges weakly\(^*\) in \(L^\infty\) to the same local minimizer. These approximation properties follow under suitable refinements of the involved discretization grids. Our results use familiar concepts arising from the analytical properties of the underlying PDE and complement previous results, derived from a topology optimization point of view.

我们从整数编程的角度考虑为海姆霍兹方程描述的波浪设计披风的问题。可以将该问题建模为PDE约束的优化问题，该问题具有分布在计算域中的整数值控制输入。先离散然后优化的方法会导致大规模混合整数非线性程序，该程序通常难以处理，因为该域离散化会产生大量整数变量。取而代之的是，我们提出了一种有效的算法，该算法能够近似地求解底层非凸无限维问题的局部信息，而无需将离散有限维整数程序求解为最优。我们仅优化局部极小值的逼近的连续松弛，然后应用总和舍入方法获得整数值控件。如果离散连续松弛的解收敛到连续松弛的局部极小值，则得到的离散值控制序列将弱收敛\（^ * \）在\（L ^ \ infty \）到相同的局部极小。这些近似特性遵循所涉及的离散化网格的适当改进。我们的结果使用了从基础PDE的分析属性中得出的熟悉概念，并补充了从拓扑优化角度得出的先前结果。