The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in the literature. Previous algorithmic solutions were typically nonconvex heuristics and were often developed in a case-by-case manner for specific structured affine spaces. In this short note we describe a general family of convex relaxations for the problem by reformulating it as a question of checking feasibility of a system of polynomial equations, and then leveraging tools from the optimization literature to obtain semidefinite programming relaxations. Our system of polynomial equations may be viewed as a matricial analog of polynomial reformulations of 0/1 combinatorial optimization problems, for which semidefinite relaxations have been extensively investigated. We illustrate numerically the utility of our approach in stylized examples that are drawn from various applications.

仿射逆特征值问题包括在仿射空间中识别具有一组预定特征值的实对称矩阵。由于其在应用中的普遍性，在文献中对该问题的各种实例进行了广泛的研究。先前的算法解决方案通常是非凸启发式算法，并且通常针对特定的结构化仿射空间以个案的方式进行开发。在此简短说明中，我们通过将其重新格式化为检查多项式方程组的可行性的问题，然后利用优化文献中的工具来获得半定规划松弛，来描述该问题的一般凸松弛族。我们的多项式方程组可以看作是0/1组合优化问题的多项式重构的矩阵模拟，对于半确定松弛，已经进行了广泛的研究。我们在从各种应用程序中提取的风格化示例中，以数字方式说明了我们的方法的效用。