Semidefinite Characterization of Sum-of-Squares Cones in Algebras

Abstract

We extend Nesterov's semidefinite programming characterization of squared functional systems, and Faybusovich's abstraction to bilinear symmetric maps, to cones of sum-of-squares elements in general abstract algebras. Using algebraic techniques such as isomorphism, linear isomorphism, tensor products, sums, and direct sums, we show that many concrete cones are in fact sum-of-squares cones with respect to some algebra and thus are representable by the cone of positive semidefinite matrices. We also consider nonnegativity with respect to a proper cone $\mathcal{K}$ and show that in some cases cones of $\mathcal{K}$-nonnegative functions are either sum of squares or at least semidefinite representable. For example, we show that some well-known Chebyshev systems, when extended to Euclidean Jordan algebras, induce cones that are semidefinite representable. Finally we will discuss some concrete examples and applications, including minimum ellipsoid enclosing given space curves, minimization of eigenvalues of polynomial matrix pencils, approximation of functions by shape-constrained functions, and approximation of combinatorial optimization problems by polynomial programming.