Semidefinite Characterization of Sum-of-Squares Cones in Algebras

We extend Nesterov’s semidefinite programming characterization of squared functional systems 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 representable by the cone of positive semidefinite matrices. We also consider nonnegativity with respect to a proper cone K, and show that in some cases K-nonnegative cones are either sum-of-squares, or are semidefinite representable. For example we show that some well-known Chebyshev systems, when extended to Euclidean Jordan algebras, induce cones that are either Sum-of-Squares cones or 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. Acknowledgements: The authors gratefully acknowledge support of the National Science Foundation through Grant # NSF-CMMI-0935305.