Cone-Constrained Singular Value Problems

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.