We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart--Thomas functions which is known from literature for certain convex problems. Since we have an integrality constraint, the previous analysis from Caillaud and Chambolle [10] does not hold anymore. Even weaker $\Gamma$-convergence results do not hold anymore because the recovery sequences generally need to attain non-integer values to recover the total variation of the limit function. We solve this issue by introducing a discretization of the input functions on an embedded, finer mesh. A superlinear coupling of the mesh sizes implies an averaging on the coarser mesh of the Raviart--Thomas ansatz, which enables to recover the total variation of integer-valued limit functions with integer-valued discretized input functions. Moreover, we are able to estimate the discretized total variation of the recovery sequence by the total variation of its limit and an error depending on the mesh size ratio. For the discretized optimization problems, we additionally add a constraint that vanishes in the limit and enforces compactness of the sequence of minimizers, which yields their convergence to a minimizer of the original problem. This constraint contains a degree of freedom whose admissible range is determined. Its choice may have a strong impact on the solutions in practice as we demonstrate with an example from imaging.

翻译：我们引入了带有总变分正则化及优化变量整性约束的无限维优化问题的离散化方法。针对已知文献中用于特定凸问题的Raviart-Thomas函数，我们推进了总变分项对偶形式的离散化。由于存在整性约束，Caillaud与Chambolle[10]先前提出的分析不再适用。更弱的Γ-收敛结果也不再成立，因为恢复序列通常需要取非整数值才能恢复极限函数的总变分。为解决该问题，我们提出在嵌入的更细网格上对输入函数进行离散化。网格尺寸的超线性耦合实现了Raviart-Thomas试探函数在粗网格上的平均化，从而能够通过取整数值的离散化输入函数恢复取整数值的极限函数的总变分。此外，我们能够通过极限函数的总变分以及依赖于网格尺寸比的误差来估计恢复序列的离散化总变分。对于离散化优化问题，我们额外添加了一个在极限情况下消失且迫使极小化序列紧致的约束条件，从而保证序列收敛至原问题的一个极小解。该约束包含一个自由参数，其允许范围得以确定。如成像实例所示，该参数的选择在实践中可能对解产生显著影响。