A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover, it enables the application of fully corrective generalized conditional gradient methods for their efficient solution. In this manuscript, this program is adapted to the minimization of a smooth convex fidelity term which is augmented with an unbalanced transport regularization term given in the form of a generalized Kantorovich–Rubinstein norm for Radon measures. More precisely, we show that the extremal points associated to the latter are given by all Dirac delta functionals supported in the spatial domain as well as certain dipoles, i.e., pairs of Diracs with the same mass but with different signs. Subsequently, this characterization is used to derive precise first-order optimality conditions as well as an efficient solution algorithm for which linear convergence is proved under natural assumptions. This behavior is also reflected in numerical examples for a model problem.

非平滑惩罚子级集极值点的精确表征既提供了有关最小化器的详细信息，也提供了涉及它们的一般类最小化问题的最优性条件。此外，它还可以应用完全校正的广义条件梯度方法来实现高效的解决方案。在本手稿中，该程序适用于平滑凸保真度项的最小化，该保真度项通过以氡测量的广义坎托罗维奇-鲁宾斯坦范数的形式给出的不平衡传输正则化项进行增强。更准确地说，我们表明与后者相关的极值点由空间域中支持的所有狄拉克δ泛函以及某些偶极子（即质量相同但符号不同的狄拉克对）给出。随后，该表征用于推导精确的一阶最优性条件以及有效的求解算法，在自然假设下证明线性收敛。这种行为也反映在模型问题的数值示例中。