Abstract: We considered a class of sum of squares (SOS) convex polynomial optimization problems with spectrahedral uncertainty data in both objective and constraint functions. Firstly, an alternative theorem for SOS-convex polynomial system with uncertain data was established in terms of SOS conditions. Secondly, we introduced a SOS relaxation dual problem for this SOS polynomial optimization problem and characterized the robust weak and strong duality properties between them. Finally, a numerical example was used to demonstrate that the SOS relaxation dual problem could be reformulated as a semidefinite  programming problem.

Key words: SOS-convex polynomial, robust duality, alternative theorem