Computing cone-constrained singular values of matrices

TL;DR

This paper studies cone-constrained singular values of a matrix $A$ relative to closed convex cones $P$ and $Q$, introducing ${\rm SV}(A,P,Q)$, ${\rm MA}(P,Q)$, and ${\rm PSV}(A)$. It establishes NP-hardness for these problems and identifies polynomial-time solvable instances, then proposes four algorithms—two exact (BFAS and a Gurobi-based solver) and two heuristics (E-AO and SRPL)—to compute cone-constrained values. The authors demonstrate the methods on diverse settings, including the Schur cone, cone pairs with nonnegative orthants, and circulant matrix families, and connect ${\rm PSV}$ to biclique problems and maximal cone angles. The work provides a practical toolbox for cone-constrained spectral problems, with implications for spectral graph theory and related optimization tasks, and highlights when efficient computation is feasible. The results offer both theoretical hardness boundaries and actionable algorithms for applications in graph analysis and matrix factorization under cone constraints.

Abstract

The concept of singular values of a rectangular matrix $A$ relative to a pair of closed convex cones $(P,Q)$ has been recently introduced by Seeger and Sossa (Cone-constrained singular value problems, Journal of Convex Analysis 30, pp. 1285-1306, 2023). These singular values are the critical (stationary) values of the non-convex optimization problem of minimizing $\langle u,Av\rangle$ such that $u$ and $v$ are unit vectors in $P$ and $Q$, respectively. When $A$ is the identity matrix, the singular values coincide with the cosine of the critical angles between $P$ and $Q$. When $P$ and $Q$ are positive orthants, the singular values are called Pareto singular values of $A$ and have applications, for instance, in spectral graph theory. This paper deals with the numerical computation of these cone-constrained singular values. We prove the NP-hardness of all the above problems, while identifying cases when such problems can be solved in polynomial time. We then propose four algorithms. Two are exact algorithms, meaning that they are guaranteed to compute a globally optimal solution; one uses an exact non-convex quadratic programming solver, and the other a brute-force active-set method. The other two are heuristics, meaning that they rapidly compute locally optimal solutions; one uses an alternating projection algorithm with extrapolation, and the other a sequential partial linearization approach based on fractional programming. We illustrate the use of these algorithms on several examples.

Figures (3)

• Figure 1: Illustration of the McCormick envelope for the nonlinear constraint $w=u,v$, with $u \in [-1,1]$ and $v \in [-1,1]$. We only display the first hyperplane that provides an overapproximation of $w$ on the domain, that is, $w = uv \leq -u + v +1$ for $-1 \leq u,v \leq 1$.
• Figure 2: E-AO, SRPL, BFAS and Gurobi compared on the problem of finding the maximum angle between the Schur cone and the nonnegative orthant (left image), and between the Schur cone and itself (right image) in dimension $n=200$. For E-AO and SRPL, $100$ iterations from random generated points are plotted in lighter colors, and their average with a thick line. The x-axis represents time in seconds.
• Figure 4: First column: Largest angle between block-circulant matrices in $\mathcal{P}_n$ and in $\mathcal{N}_n$. Second column: Largest known angle between $\mathcal{P}_n$ and $\mathcal{N}_n$.

Theorems & Definitions (34)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• Corollary 1
• Lemma 1
• proof
• Theorem 3
• proof
• Remark 1: Further connection between ${\rm SV}(A,P,Q)$ and ${\rm MA}(\widetilde{P},\widetilde{Q})$