We extend Nesterov's semidefinite programming (SDP) characterization of the cone of functions that can be expressed as sums of squares (SOS) of functions in finite dimensional linear functional spaces. Our extension is to algebraic systems that are endowed with a binary operation which map two elements of a finite dimensional vector space to another vector space; the binary operation must follow the distributive laws. We derive a number of previously known SOS characterizations as a special case of our framework. In addition to Nesterov's result (Nesterov, 2000) for finite dimensional linear functional spaces, we show that the cone of positive semidefinite univariate polynomials with symmetric matrices as coefficients, SOS polynomials with coefficients from Euclidean Jordan algebras (first studied by Kojima and Muramatsu, 2007), and numerous other problems involving vector-valued functions not previously considered can be expressed in our framework. Some potential applications in geometric design problems with constraints on curvature of space curves, and in multivariate approximation theory problems with convexity as constraint are briefly discussed.
Citation
RUTCOR Research Report RRR 27-2009, Rutgers Center for Operations Research, Rutgers University, NJ, December 2009