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The Lasserre hierarchy of semidefinite programming (SDP) relaxations is aneffective scheme for finding computationally feasible SDP approximations ofpolynomial optimization over compact semi-algebraic sets. In this paper, weshow that, for convex polynomial optimization, the Lasserre hierarchy with aslightly extended quadratic module always converges asymptotically even in theface of non-compact semi-algebraic feasible sets. We do this by exploiting acoercivity property of convex polynomials that are bounded below. We furtherestablish that the positive definiteness of the Hessian of the associatedLagrangian at a saddle-point (rather than the objective function at eachminimizer) guarantees finite convergence of the hierarchy. We obtain finiteconvergence by first establishing a new sum-of-squares polynomialrepresentation of convex polynomials over convex semi-algebraic sets under asaddle-point condition. We finally prove that the existence of a saddle-pointof the Lagrangian for a convex polynomial program is also necessary for thehierarchy to have finite convergence.

Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness