We consider semidefinite programming problems on which a permutation group is acting. We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order matrix *-representation of the commutant (centralizer ring) of the matrix algebra generated by the permutation matrices. We apply it to… (More)
A standard quadratic problem consists of nding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semideenite programming relaxation is strengthened by replacing the cone of positive semideenite matrices by the cone of completely positive matrices (the positive semideenite matrices which allow a factorization F F T… (More)
We consider the problem of computing the minimum value p min taken by a polynomial p(x) of degree d over the standard simplex ∆. This is an NP-hard problem already for degree d = 2. For any integer k ≥ 1, by minimizing p(x) over the set of rational points in ∆ with denominator k, one obtains a hierarchy of upper bounds p ∆(k) converging to p min as k −→ ∞.… (More)
We consider the problem of global minimization of rational functions on IR n (unconstrained case), and on an open, connected, semi-algebraic subset of IR n , or the (partial) closure of such a set (constrained case). We show that in the univariate case (n = 1), these problems have exact reformulations as semidefinite programming (SDP) problems, by using… (More)
It has been long conjectured that the crossing numbers of the complete bipartite graph K m,n and of the complete graph K n equal Z(m, n) := n 2 n−1 2 m 2 m−1 2 and Z(n) := 1 4 n 2 n−1 2 n−2 2 n−3 2 , respectively. In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of… (More)
We consider semidefinite programming relaxations of the quadratic assignment problem, and show how to exploit group symmetry in the problem data. Thus we are able to compute the best known lower bounds for several instances of quadratic assignment problems from the problem library: [R.
It has been long–conjectured that the crossing number cr(K m,n) of the complete bi-partite graph K m,n equals the Zarankiewicz Number Z(m, n) := ⌊ m−1 2 ⌋⌊ m 2 ⌋⌊ n−1 2 ⌋⌊ n 2 ⌋. Another long–standing conjecture states that the crossing number cr(K n) of the complete graph K n equals Z(n) := 1 4 n 2 n−1 2 n−2 2 n−3 2. In this paper we show the following… (More)
It was recently shown in  that, unlike in linear optimization, the central path in semidefinite optimization (SDO) does not converge to the analytic center of the optimal set in general. In this paper we analyze the limiting behavior of the central path to explain this unexpected phenomenon. This is done by deriving a new necessary and sufficient… (More)