SIAM Review, Volume 63, Issue 4, Page 783-821, January 2021.
We consider a class of infinite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given finite set $\mathcal{M}\subset\mathbb{R}^m$. Such hybrid discrete-continuous problems occur in, e.g., topology optimization or medical imaging and are challenging due to their lack of weak lower semicontinuity. To circumvent this difficulty, we introduce as a regularization term a convex integral functional with an integrand that has a polyhedral epigraph with vertices corresponding to the values of $\mathcal{M}$; similar to the $L^1$ norm in sparse regularization, this “vector multibang penalty” promotes solutions with the desired structure while allowing the use of tools from convex optimization for the analysis as well as the numerical solution of the resulting problem. We show well-posedness of the regularized problem and analyze stability properties of its solution in a general setting. We then illustrate the approach for three specific model optimization problems of broader interest: optimal control of the Bloch equation, optimal control of an elastic deformation, and a multimaterial branched transport problem. In the first two cases, we derive explicit characterizations of the penalty and its generalized derivatives for a concrete class of sets $\mathcal{M}$. For the third case, we discuss the algorithmic computation of these derivatives for general sets. These derivatives are then used in a superlinearly convergent semismooth Newton method applied to a sequence of regularized optimization problems. We illustrate the behavior of this approach for the three model problems with numerical examples.

中文翻译：

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离散向量值优化问题的凸松弛

SIAM 评论，第 63 卷，第 4 期，第 783-821 页，2021 年 1 月。
我们考虑一类无限维优化问题，其中分布式向量值变量几乎在任何地方都应该从给定的有限集 $\mathcal{M}\subset\mathbb{R}^m$ 中取值。这种混合离散连续问题出现在例如拓扑优化或医学成像中，并且由于它们缺乏弱的下半连续性而具有挑战性。为了规避这个困难，我们引入了一个凸积分函数作为正则化项，其被积函数具有一个多面体题词，顶点对应于 $\mathcal{M}$ 的值；类似于稀疏正则化中的 $L^1$ 范数，这种“向量多重惩罚”促进了具有所需结构的解决方案，同时允许使用来自凸优化的工具进行分析以及结果问题的数值解决方案。我们展示了正则化问题的适定性，并在一般情况下分析了其解的稳定性属性。然后，我们说明了三个更广泛感兴趣的特定模型优化问题的方法：布洛赫方程的最优控制、弹性变形的最优控制和多材料分支传输问题。在前两种情况下，我们为一组具体的集合 $\mathcal{M}$ 推导出惩罚及其广义导数的显式特征。对于第三种情况，我们讨论一般集合的这些导数的算法计算。然后将这些导数用于超线性收敛半光滑牛顿法，应用于一系列正则化优化问题。我们用数值例子说明了这种方法对于三个模型问题的行为。