Abstract Cones of Positive Polynomials and Their Sums of Squares Relaxations

Roland Hildebrand (1)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques

We present a new family of sums of squares (SOS) relaxations to cones of positive polynomials. The SOS relaxations employed in the literature are cones of polynomials which can be represented as ratios, with an SOS as numerator and a fixed positive polynomial as denominator. We employ nonlinear transformations of the arguments instead. A fixed cone of positive polynomials, considered as a subset in an abstract coefficient space, corresponds to an infinite, partially ordered set of concrete cones of positive polynomials of different degrees and in a different number of variables. To each such concrete cone corresponds its own SOS cone, leading to a hierarchy of increasingly tighter SOS relaxations for the abstract cone.

Domaines: Optimisation et contrôle [math.OC]

Roland Hildebrand. Abstract Cones of Positive Polynomials and Their Sums of Squares Relaxations. Moritz Diehl, FRançois Glineur, Elias Jarlebring, Wim Michiels. Recent Advances in Optimization and its Applications in Engineering. The 14th Belgian-French-German Conference on Optimization, Springer, pp.41-50, 2010, 978-3-642-12597-3. ⟨10.1007/978-3-642-12598-0_4⟩. ⟨hal-00765814⟩