SIAM Journal on Optimization, Volume 30, Issue 3, Page 1922-1953, January 2020.
In general, standard necessary optimality conditions cannot be formulated in a straightforward manner for semismooth shape optimization problems. In this paper, we consider shape optimization problems constrained by variational inequalities of the first kind, so-called obstacle-type problems. Under appropriate assumptions, we prove existence of adjoints for regularized problems and convergence to adjoints of the unregularized problem. Moreover, we derive shape derivatives for the regularized problem and prove convergence to a limit object. Based on this analysis, an efficient optimization algorithm is devised and tested numerically.

中文翻译：

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使用伴随变量的变分不等式形状优化的高效技术

SIAM优化杂志，第30卷，第3期，第1922-1953页，2020
年1月。一般而言，对于半光滑形状优化问题，无法以简单的方式制定标准必要的最优条件。在本文中，我们考虑了由第一类变分不等式约束的形状优化问题，即所谓的障碍型问题。在适当的假设下，我们证明了正则化问题的伴随性的存在以及未正规化问题的伴随性的收敛性。此外，我们导出了正则化问题的形状导数，并证明了收敛到极限对象。基于此分析，设计了有效的优化算法并进行了数值测试。