Parameter-Dependent Slack Variable approach for positivity check of polynomials over hyper-rectangle

This paper addresses the positivity check of polynomials in which the region of indeterminates is given as a hyper-rectangle. A new tractable quadratically parameter-dependent condition is proposed using parameter-dependent slack variables (PDSVs) which are up to second-order with respect to the indeterminates. It is proved that our derived condition always holds if the condition via sum-of-squares (SOS) approach holds. In addition, as the polynomials defining the region of the indeterminates always satisfy constraint qualification, our proposed method is proved to be a sufficient condition which asymptotically becomes a necessary and sufficient condition for our addressed problem with increase of the sizes of PDSVs.