Polynomial optimization with a focus on hyperbolic polynomials

A problem of minimizing (or maximizing) a multivariate polynomial over a subset of an Euclidean Space defined by the solution set of finitely many polynomial equations and inequalities is called a Polynomial Optimization Problem (PoP). PoPs describe a class of optimization problems with a nontrivial amount of geometric, algebraic and analytic properties. At the same time, POPs are general enough to capture as a special case, a very wide swath of finite dimensional optimization problems and even some semi-infinite optimization problems. We can reformulate PoPs in many equivalent forms. For example, by introducing a new variable (increasing the dimension of the space by one), we can push the objective function’s multivariate polynomial into the constraints (hence, without loss of generality, we may assume that the objective function is always linear). In this setting, an interesting, nicely structured, and powerful class of convex PoPs is Semidefinite Programming (SDP) problems. Objective functions of SDPs are linear, and their feasible solution sets are defined as the intersection of the convex cone of n-by-n symmetric positive semidefinite matrices (all real entries), denoted S+ with an affine subspace. Such convex sets are called spectrahedra. In SDP problems we may also use additional auxiliary variables that effectively get projected away due to our carefully picked choices for the objective function of the SDP. This observation shows that SDPs can also deal with convex sets that are orthogonal projections of spectrahedra. These latter convex sets are called spectrahedral shadows. Indeed, spectrahedral shadows yield a strict superset of spectrahedra. However, except for utilization of facial exposedness property (of spectrahedra), we do not have many elegant, useful certificates helping us distinguish these two families of convex sets precisely (see [25, 8, 23, 20, 3, 16]). We consider PoPs from a convex optimization viewpoint (see for instance [9, 11, 19]). Then a central question is “when is the feasible region of a PoP convex? ” This leads us to hyperbolic polynomials (a.k.a. stable polynomials, under a suitable transformation) which naturally define convex domains. For the sake of convenience, we work with homogeneous hyperbolic polynomials so that the underlying convex domains become convex cones called hyperbolicity cones. Let p be a homogeneous polynomial (this is without loss of generality in our current context) of degree d in n variables, and let e ∈ R. p is said to be hyperbolic in direction e if p(e) > 0 and, for all x ∈ R, the scalar polynomial λ 7→ p(x− λe) has only real roots. Studies of hyperbolic polynomials go back at least to the work of Petrovsky (from the 1930s). Considerable amount of work has been done by G̊arding, Atiyah, Bott and G̊arding as well as Hörmander. Since the early 1990’s there has been an amazing amount of activity allowing the subject to branch into systems and control theory, operator theory (see Marcus-Spielman-Srivastava [13] solution of Kadison-Singer problem) interior-point methods (see, for instance, [5, 20, 15] and the references therein), discrete optimization and combinatorics (see, for instance, Gurvits [4], Wagner [27] and references therein) semidefinite programming