On the Convexification of Spectral Sets Induced by Non-Invariant Sets

Given a finite-dimensional FTvN system $(\mathbb{V},\mathbb{W},λ)$, we study the convexification of the spectral set $λ^{-1}(\mathcal{C})$ induced by a set $\mathcal{C} \subseteq \mathbb{W}$. While the case of invariant $\mathcal{C}$ has been relatively well-studied, the results for non-invariant $\mathcal{C}$ are largely lacking in the literature. We fill this void by developing simple and geometric characterizations of the convex hull and closed convex hull of $λ^{-1}(\mathcal{C})$ when $\mathcal{C}$ has no invariance property. We further specialize our results to the case of invariant $\mathcal{C}$, and obtain new convexifications of $λ^{-1}(\mathcal{C})$ in this case.

💡 Research Summary

This paper investigates the convexification of spectral sets in finite‑dimensional FTvN (Freudenthal‑Tits‑von Neumann) systems, extending the theory beyond the previously studied case where the underlying set C is invariant under the spectral map λ. An FTvN system consists of two inner‑product spaces V and W and a nonlinear map λ:V→W that satisfies (P1) an inner‑product domination property and (P2) a majorization property. From these axioms it follows that λ is norm‑preserving, its range K=ran λ is a closed convex cone, and λ is 1‑Lipschitz.

The authors focus on the set E=λ⁻¹(C) for an arbitrary feasible set C⊆W (i.e., C∩K≠∅). Earlier works showed that when C is invariant—meaning there exists a reduced system (W,W,μ) with C=μ⁻¹(Q)—the “transfer principle” holds:  clconv λ⁻¹(C)=λ⁻¹(clconv C) and conv λ⁻¹(C)=λ⁻¹(conv C). However, many practical optimization problems involve non‑invariant sets such as general polyhedra or sparse ellipsoids, for which no convexification results were available.

The core contribution is a pair of geometric characterizations that rely only on two objects: the convex hull of C∩K and the polar cone K° of K. Lemma 3.1 establishes that for any single vector u∈K,  conv λ⁻¹(u)=λ⁻¹(u+K°), which is equivalent to the majorization relation x≺u. Building on this, Theorem 3.1 proves that for any non‑empty convex D⊆K,  conv λ⁻¹(D)=λ⁻¹(D+K°)= { x∈V | ∃ u∈D with λ(x)−u∈K° }. If D+K° is closed, the same expression gives the closed convex hull.

The main result, Theorem 3.2, lifts the restriction D⊆K by applying the previous theorem to the feasible set C. It shows that  conv λ⁻¹(C)=λ⁻¹( conv(C∩K) + K° ), and for any convex set D satisfying conv(C∩K)⊆D⊆clconv(C∩K),  clconv λ⁻¹(C)=cl λ⁻¹(D+K°). When D+K° is closed (e.g., if D is compact or polyhedral), the closure can be omitted. The theorem also clarifies that one cannot simply replace conv(C∩K) by (conv C)∩K in general; counter‑examples are provided.

The paper supplies concrete examples. In the simplest case λ(x)=x↓ on ℝ², the cone K is the non‑increasing orthant and K° is the line {(−α,α):α≥0}. For a two‑point set C, the convex hull of λ⁻¹(C) is the standard 2‑simplex, matching the formula. A more involved example uses the singular‑value map (E2). Here K is the decreasing non‑negative orthant, K° is the set of vectors with non‑positive cumulative sums, and C is a k‑sparse ellipsoid defined by a positive‑definite matrix A. The intersection C∩K has a simple description (ordered non‑negative coordinates followed by zeros), and the theorem yields an explicit description of conv λ⁻¹(C) as a set of matrices whose singular values belong to (C∩K)+K°. This directly translates into convex relaxations for problems with sparse spectral constraints.

Section 3.2 (Theorem 3.3, not fully reproduced) strengthens the results under additional structural assumptions on C∩K (e.g., polyhedrality or compactness), guaranteeing that D+K° is closed and thus providing exact closed‑convex hull formulas without taking closures.

Finally, the authors revisit the invariant case. By specializing the general formulas, they recover and improve earlier transfer principles, removing unnecessary compactness or continuity assumptions. This yields new convexification formulas for invariant sets that hold in any FTvN system, covering eigenvalue, singular‑value, and absolute‑reordering maps as special cases.

Overall, the paper delivers a unified, geometric framework for convexifying spectral sets without relying on invariance. The results are applicable to a broad class of optimization problems where spectral constraints appear, such as low‑rank matrix recovery, eigenvalue‑constrained design, and sparse principal component analysis. The approach suggests several future directions: extending to infinite‑dimensional FTvN systems, handling time‑varying feasible sets, and developing efficient algorithms for projecting onto λ⁻¹(conv(C∩K)+K°) using the polar cone structure.