This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial optimization problems in the face of data uncertainty. The class of convex optimization problems, called robust SOS-convex polynomial optimization problems, includes robust quadratically constrained convex optimization problems and robust separable convex polynomial optimization problems. It establishes sums-of-squares polynomial representations characterizing robust solutions and exact SDP-relaxations of robust SOS-convex polynomial optimization problems under various commonly used uncertainty sets. In particular, the results show that the polytopic and ellipsoidal uncertainty sets, that allow second-order cone re-formulations of robust quadratically constrained optimization problems, continue to permit exact SDP-relaxations for a broad class of robust SOS-convex polynomial optimization problems.
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