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- Etienne de Klerk, Dmitrii V. Pasechnik, Alexander Schrijver
- Math. Program.
- 2007

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)

- Immanuel M. Bomze, Etienne de Klerk
- J. Global Optimization
- 2002

- Etienne de Klerk, Monique Laurent, Pablo A. Parrilo
- Theor. Comput. Sci.
- 2006

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)

- Etienne de Klerk, Renata Sotirov
- Math. Program.
- 2010

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.

- Immanuel M. Bomze, Mirjam Dür, Etienne de Klerk, Kees Roos, Arie J. Quist, Tamás Terlaky
- J. Global Optimization
- 2000

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)

- Etienne de Klerk, John Maharry, Dmitrii V. Pasechnik, R. Bruce Richter, Gelasio Salazar
- SIAM J. Discrete Math.
- 2006

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)

- Dorina Jibetean, Etienne de Klerk
- Math. Program.
- 2006

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 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)

The problem of colouring a k-colourable graph is well-known to be NP-complete, for k ≥ 3. The MAX-k-CUT approach to approximate k-colouring is to assign k colours to all of the vertices in polynomial time such that the fraction of 'defect edges' (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and… (More)