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：本文针对具有全变差正则化项及优化变量整性约束的无限维优化问题，提出了相应的离散化方法。我们推进了全变差项对偶公式的离散化技术，该技术采用Raviart-Thomas函数，在特定凸问题研究中已有文献记载。由于整性约束的存在，Caillaud与Chambolle[10]先前提出的分析方法不再适用。甚至更弱的$\Gamma$-收敛结论也不再成立，因为恢复序列通常需要取非整数值才能逼近极限函数的全变差。为解决此问题，我们通过在嵌入式细网格上对输入函数进行离散化。网格尺寸的超线性耦合使得Raviart-Thomas格式在粗网格上产生平均效应，从而能够用整数值离散输入函数恢复整数值极限函数的全变差。此外，我们能够通过极限函数的全变差及依赖于网格尺寸比的误差项来估计恢复序列的离散化全变差。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的约束条件，该条件保证了极小化序列的紧致性，从而使其收敛于原问题的极小解。该约束包含一个自由度，我们确定了其允许取值范围。如我们在成像示例中所展示的，该自由度的选择可能对实际解产生显著影响。