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Journal of Convex Analysis 30 (2023), No. 4, 1285--1306
Copyright Heldermann Verlag 2023

Cone-Constrained Singular Value Problems

Alberto Seeger
Dept. of Mathematics, University of Avignon, France

David Sossa
Instituto de Ciencias de la Ingenierķa, Universidad de O'Higgins, Rancagua, Chile


The singular values of a matrix $A$ of size $m\times n$ can be seen as the critical values of the bilinear form $\langle u,Av\rangle$ with $u$ and $v$ ranging over the unit spheres of $\mathbb{R}^m$ and $\mathbb{R}^n$, respectively. If $u$ and $v$ are further restricted by closed convex cones $P$ and $Q$, respectively, then the criticality conditions are: $P\ni u \perp (Av-\sigma u)\in P^\ast$, $Q\ni v \perp (A^\top u -\sigma v)\in Q^\ast$. This is a coupled system of complementarity problems involving a pair of cones and their dual cones. The parameter $\sigma$ is called a singular value of $A$ relative to $(P,Q)$. The purpose of our work is to study this new concept of singular value. The analysis of such a coupled system is motivated by a number of applications. By way of illustration, we consider a nonnegative Principal Component Analysis problem.

Keywords: Convex cone, cone-constrained singular value, cone-constrained eigenvalue, complementarity problem, principal component analysis.

MSC: 15A18, 52A40, 90C26, 90C33.

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