Papers › Computing cone-constrained singular values of matrices
Computing cone-constrained singular values of matrices
Giovanni Barbarino, Nicolas Gillis, David Sossa
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
This paper deals with the numerical computation of the least singular value of a rectangular matrix A relative to a pair of closed convex cones (P,Q), which is defined as the optimal value of the non-convex optimization problem of minimizing ⟨u,Av⟩ such that u and v are unit vectors in P and Q, respectively. When A is the identity matrix, the least singular value coincides with the cosine of the largest angle between P and Q. When P and Q are positive orthants, the least singular value is called the least Pareto singular value of A and has applications, for instance, in graph theory. 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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections