Convergent SDP-Relaxations for Polynomial Optimization with Sparsity

We consider a polynomial programming problem P on a compact semi-algebraic set K ⊂ Rn, described by m polynomial inequalities gj(X) ≥ 0, and with criterion f ∈ R[X]. We propose a hierarchy of semidefinite relaxations in the spirit those of Waki et al. [9]. In particular, the SDP-relaxation of order r has the following two features: (a) The number of variables is O(κ2r) where κ = max[κ1, κ2] witth κ1 (resp. κ2) being the maximum number of variables appearing the monomials of f (resp. appearing in a single constraint gj(X) ≥ 0). (b) The largest size of the LMI’s (Linear Matrix Inequalities) is O(κr). This is to compare with the respective number of variables O(n2r) and LMI size O(nr) in the original SDP-relaxations defined in [11]. Therefore, great computational savings are expected in case of sparsity in the data {gj , f}, i.e. when κ is small, a frequent case in practical applications of interest. The novelty with respect to [9] is that we prove convergence to the global optimum of P when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory. In such cases, and as a by-product, we also obtain a new representation result for polynomials positive on a basic closed semialgebraic set, a sparse version of Putinar’s Positivstellensatz [16].