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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)

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 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.

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)

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)

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)

We derive a semidefinite relaxation of the satisfiability (SAT) problem and discuss its strength. We give both the primal and dual formulation of the relaxation. The primal formulation is an eigenvalue optimization problem, while the dual formulation is a semidefinite feasibility problem. It is shown that using the relaxation, the notorious pigeon hole and… (More)

We provide a monotone non increasing sequence of upper bounds f H k (k ≥ 1) converging to the global minimum of a polynomial f on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper bounds in [J.B. Lasserre, A new look at nonnegativity on closed sets and polynomial optimization, SIAM J. Optim. 21, pp. 864–885,… (More)

- ETIENNE DE KLERK
- 2008

We consider a new semidefinite programming (SDP) relaxation of the symmetric traveling salesman problem (TSP), that may be obtained via an SDP relaxation of the more general quadratic assignment problem (QAP). We show that the new relaxation dominates the one in the paper: Unlike the bound of Cvetkovi´c et al., the new SDP bound is not dominated by the… (More)