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Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness
The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an
effective scheme for finding computationally feasible SDP approximations of
polynomial optimization over compact semi-algebraic sets. In this paper, we
show that, for convex polynomial optimization, the Lasserre hierarchy with a
slightly extended quadratic module always converges asymptotically even in the
face of non-compact semi-algebraic feasible sets. We do this by exploiting a
coercivity property of convex polynomials that are bounded below. We further
establish that the positive definiteness of the Hessian of the associated
Lagrangian at a saddle-point (rather than the objective function at each
minimizer) guarantees finite convergence of the hierarchy. We obtain finite
convergence by first establishing a new sum-of-squares polynomial
representation of convex polynomials over convex semi-algebraic sets under a
saddle-point condition. We finally prove that the existence of a saddle-point
of the Lagrangian for a convex polynomial program is also necessary for the
hierarchy to have finite convergence.Comment: 17 page
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