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A Lagrangian relaxation view of linear and semidefinite hierarchies

We consider the general polynomial optimization problem $P: f^*=\min \{f(x)\,:\,x\in K\}$ 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 $\tilde{P}$ 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 in which a nonnegative polynomial (with coefficients to be determined) is replaced with a constant polynomial equal to zero. Inspired by this interpretation, 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 parametrized hierarchy, 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. The parameter $k$ is to be decided by the user. When applied to a non trivial class of convex problems, the first relaxation of the parametrized hierarchy is exact, in contrast with the LP-hierarchy where convergence cannot be finite. When applied to 0/1 programs it is at least as good as the first one in the hierarchy of semidefinite relaxations. However obstructions to exactness still exist and are briefly analyzed. Finally, the standard semidefinite hierarchy can also be viewed as a simplification of an extended Lagrangian relaxation, but different in spirit as sums of squares (and not scalars) multipliers are allowed.