Abstract:

While one can easily characterize the (closure of) hyperbolicity cone $K_p$ associated to an arbitrary hyperbolic polynomial $p$ in terms of finitely many polynomial inequalities and construct a logarithmic self-concordant barrier (SCB) functional for $K_p$, little is known about its dual cone $K_p^*$. Elementary symmetric polynomials can be thought of as derivative polynomials (in a certain sense) of $E_n(x)=\prod_{i=1,\ldots,n}x_i$; their associated hyperbolicity cones give a natural sequence of relaxations for $\Re^n_+=K_{E_n}$. Following the establishment of the recursive structure for these cones, we give an algebraic characterization for the dual cone associated with $E_{n-1}(x)=\sum_{1\leq i\leq n}\prod_{j\neq i}x_j$ which was previously unknown and show how one can easily construct a SCB functional for this cone.