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.
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The authors were supported in part by NSF Grants CCF-1350590 and CCF-1637598, by AFOSR Grant FA9550-16-1-0210, and by a Sloan research fellowship.
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Candogan, U., Soh, Y.S. & Chandrasekeran, V. A note on convex relaxations for the inverse eigenvalue problem. Optim Lett 15, 2757–2772 (2021). https://doi.org/10.1007/s11590-021-01708-1
- Combinatorial optimization
- Real algebraic geometry
- Schur–Horn orbitope
- Semidefinite programming
- Sums of squares polynomials