On the Projection-Based Convexification of Some Spectral Sets

Abstract: Given a finite-dimensional real inner-product space $\mathbb{E}$ and a closed convex cone $\mathcal{K}\subseteq\mathbb{R}^n$, we call $\lambda:\mathbb{E}\to\mathcal{K}$ a spectral map if $(\mathbb{E},\mathbb{R}^n,\lambda)$ forms a generalized Fan-Theobald-von Neumann (FTvN) system (Gowda, 2019). Common examples of $\lambda$ include the eigenvalue map, the singular-value map and the characteristic map of complete and isometric hyperbolic polynomials. We call $\mathcal{S} \subseteq \mathbb{E}$ a spectral set if $\mathcal{S} := \lambda^{-1}(\mathcal{C})$ for some $\mathcal{C}\subseteq \mathbb{R}^n$. We provide projection-based characterizations of $\mathsf{clconv}\mathcal{S}$ (i.e., the closed convex hull of $\mathcal{S}$) under two settings, namely, when $\mathcal{C}$ has no invariance property and when $\mathcal{C}$ has certain invariance properties. In the former setting, our approach is based on characterizing the bi-polar set of $\mathcal{S}$, which allows us to judiciously exploit the properties of $\lambda$ via convex dualities. In the latter setting, our results complement the existing characterization of $\mathsf{clconv}\mathcal{S}$ in Jeong and Gowda (2023), and unify and extend the related results in Kim et al. (2022) established for certain special cases of $\lambda$ and $\mathcal{C}$.