Abstract

Let A be a real matrix of size m× n . In classical linear algebra, a real number σ is called a singular value of A if there exist unit vectors u∈ Rm and v∈ Rn such that Av= σu and A⊤u= σv . In variational analysis, a singular value of A is viewed as a critical value of the bilinear form ⟨ u, Av⟩ when u and v range on the unit spheres of Rm and Rn , respectively. If u and v are further required to be nonnegative, then the idea of criticality is expressed by means of a pair of complementarity problems, namely, 0 ≤ u⊥ (Av- σu) ≥ 0 and 0 ≤ v⊥ (A⊤u- σv) ≥ 0 . The parameter σ is now called a Pareto singular value of A. In this work we study the concept of Pareto singular value and, by way of application, we analyze a problem of nonnegative matrix factorization. The set Ξ (A) of Pareto singular values of A is nonempty and finite. We derive an explicit formula for the maximum number of Pareto singular values in a matrix of prescribed size. The elements of Ξ (A) can be found by solving a collection of classical singular value problems involving the principal submatrices of A. Unfortunately, such a method is cost prohibitive if m and n are large. For matrices of large size we propose an algorithm of alternating minimization type. This work is a continuation of our paper entitled Cone-constrained singular value problems published in the Journal of Convex Analysis (30, 2023, pp. 1285–1306).