We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ(G), to a new graph parameter Ψβ(G), nested between α(G) and χ(G); Ψβ(G) is polynomial time computable if β(G) is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number χ ∗ (·) and χ(·) unless P = NP. Moreover, based on the Motzkin–Straus formulation for α(G), we give (quadratically constrained) quadratic and copositive programming formulations for χ(G). Under some mild assumptions, n/β(G) ≤ Ψβ(G), but, while n/β(G) remains below χ ∗ (G), Ψβ(G) can reach χ(G) (e.g., for β(·) =α(·)). We also define new polynomial time computable lower bounds for χ(G), improving the classic Lovász theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the follow-up paper [N. Gvozdenović and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592–615].

A common method, originally introduced by Lovász in 1979, for calculating an upper bound on the size of a maximum independent set for a graph is to consider a relaxation of the problem expressed as a semidefinite program (SDP). Today, the most prevalent method for solving a general SDP is with a primal-dual interior point method (IPM). These methods are highly developed, provide reliable convergence, and parallelize relatively well on a shared memory architecture. However, they are severely limited by their memory requirements, which grow with the square of the number of edges in the graph. Here, we investigate the boundary point method (BPM) developed by Povh, Rendl, and Weigele in 2006. Storage for this method grows as the square of the number of nodes in the graph, allowing us to bound much larger graphs. We have implemented the boundary point method in C within a branch-and-bound framework and discuss several methods used within that framework aimed at increasing the efficiency of the algorithm. We also compare the BPM with