A Note on Convex Relaxations for the Inverse Eigenvalue Problem

TL;DR

This paper introduces a general convex relaxation approach for the inverse eigenvalue problem, transforming it into a polynomial feasibility problem and applying semidefinite programming techniques.

Contribution

It presents a novel, unified convex relaxation framework for the inverse eigenvalue problem using polynomial reformulations and semidefinite programming.

Findings

Numerical examples demonstrate the effectiveness of the proposed relaxations.

The approach generalizes previous case-specific heuristics.

Semidefinite relaxations provide feasible solutions in stylized applications.

Abstract

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.