## Convergent SDP-Relaxations in Polynomial Optimization with Sparsity

Venue: | SIAM Journal on Optimization |

Citations: | 24 - 8 self |

### BibTeX

@ARTICLE{Lasserre_convergentsdp-relaxations,

author = {Jean B. Lasserre},

title = {Convergent SDP-Relaxations in Polynomial Optimization with Sparsity},

journal = {SIAM Journal on Optimization},

year = {},

volume = {17},

pages = {822--843}

}

### OpenURL

### Abstract

Abstract. We consider a polynomial programming problem P on a compact semi-algebraic set K ⊂ R n, 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(κ 2r) 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(κ r). This is to compare with the respective number of variables O(n 2r) and LMI size O(n r) 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]. 1.