A Lagrangian relaxation view of linear and semidefinite hierarchies

Consider the polynomial optimization problem P: f* = min{f(x): x ∈ K} where K is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the global optimum f*, provided that it is applied to a problem P̃ equivalent to P, in which sufficiently many redundant constraints (products of the initial ones) are added to the initial description of P. Next we show that the standard hierarchy of LP-relaxations of P (in the spirit of Sherali-Adams´ RLT) can be interpreted as a brute force simplification of the above Lagrangian relaxation. So we provide a systematic improvement of the LP-hierarchy by doing a much less brutal simplification which results into a parametrized hierarchy of semidefinite programs (and not linear programs any more). For each semidefinite program in the hierarchy parametrized by k, the semidefinite constraint has a fixed size O(n^k), independently of the rank in the hierarchy, in contrast with the standard hierarchy of semidefinite relaxations.