A note on convex relaxations for the inverse eigenvalue problem

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