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- Etienne de Klerk, Dmitrii V. Pasechnik
- SIAM Journal on Optimization
- 2002

Lovász and Schrijver showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs (SDP’s) of increasing size (lift-and-project method). In this talk we present a similar idea. We show how the stability number can be computed as the solution of a conic linear program (LP) over the cone of… (More)

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

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

Abstract. 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… (More)

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

We consider the problem of global minimization of rational functions on IR (unconstrained case), and on an open, connected, semi-algebraic subset of IR, 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)

- Etienne de Klerk, Hans van Maaren, Joost P. Warners
- Journal of Automated Reasoning
- 2000

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. We show that using the relaxation, a proof of the unsatisfiability of… (More)

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

Abstract. We consider the problem of computing the minimum value pmin 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 pmin as k… (More)

- 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
- CEJOR
- 2008

We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere. These relatively simple optimization problems have many applications. We review known approximation results as well as negative (inapproximability) results from the recent literature.

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)

- 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: [D. Cvetković, M. Cangalović and V. Kovačević-Vujčić. Semidefinite Programming… (More)