  We focus on computing certified upper bounds for the positive maximal
singular value (PMSV) of a given matrix. The PMSV problem boils down to
maximizing a quadratic polynomial on the intersection of the unit sphere and
the nonnegative orthant. We provide a hierarchy of tractable semidefinite
relaxations to approximate the value of the latter polynomial optimization
problem as closely as desired. This hierarchy is based on an extension of
P\'olya's representation theorem. Doing so, positive polynomials can be
decomposed as weighted sums of squares of $s$-nomials, where $s$ can be a
priori fixed ($s=1$ corresponds to monomials, $s=2$ corresponds to binomials,
etc.). This in turn allows us to control the size of the resulting semidefinite
relaxations.
