Continuous Optimization Spectral bounds for unconstrained (1,1)-quadratic optimization problems Walid Ben-Ameur, José Neto * Institut TELECOM, TELECOM SudParis, CNRS UMR 5157, 9 rue Charles Fourier, 91011 Evry, France article info Article history: Received 17 December 2008 Accepted 28 February 2010 Available online 7 March 2010 Keywords: Unconstrained quadratic programming Semidefinite programming Maximum cut problem abstract Given an unconstrained quadratic optimization problem in the following form: ðQPÞ minfx t Qx j x 2 f1; 1g n g; with Q 2 R nn , we present different methods for computing bounds on its optimal objective value. Some of the lower bounds introduced are shown to generally improve over the one given by a classical semidefinite relaxation. We report on theoretical results on these new bounds and provide preliminary computational experiments on small instances of the maximum cut problem illustrating their performance. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Consider a quadratic function q : R n ! R given by: qðxÞ¼ x t Qx, where Q denotes a n  n rational matrix. An unconstrained (1,1)-quadratic optimization problem can be expressed as follows: ðQPÞZ  ¼ minfqðxÞjx 2 f1; 1g n g; where f1; 1g n denotes the set of n-dimensional vectors with en- tries either equal to 1 or 1. Without loss of generality we assume that the matrix Q is symmetric. Problem ðQPÞ is a classical combinatorial optimization problem with many applications, e.g. in statistical physics and circuit design (Barahona et al., 1988; Grötschel et al., 1989; Pinter, 1984). It is well-known that any (0,1)-quadratic problem expressed as: minfx t Ax þ c t xjx 2f0; 1g n g; A 2 R nn ; c 2 R n , can be formulated in the form of problem ðQPÞ (with dimension n þ 1 instead of n) and conversely (Hammer, 1965; De Simone, 1990). Problem ðQPÞ is known to be NP-hard in general (Karp, 1972). Some polynomially solvable cases have been identified from among the following (Allemand et al., 2001; Ben-Ameur and Neto, 2008b; Çela et al., 2006). Proposition 1.1. For a fixed integer p, if the matrix Q (given by its nonzero eigenvalues and associated eigenvectors) has rank at most p and negative diagonal entries only, then problem ðQPÞ can be solved in polynomial time. Note also the following extension of the last Proposition given in Ben-Ameur and Neto (2008b). Proposition 1.2. For fixed integers p P 2 and q P 0, if the matrix Q (given by its nonzero eigenvalues and associated eigenvectors) has rank at most p and at most q positive diagonal entries, then problem ðQPÞ can be solved in polynomial time. Different methods for computing bounds for problems such as ðQPÞ have been proposed in the literature. An early reference is Hammer and Rubin (1970), in which the authors proposed a meth- od convexifying the objective function by making use of the small- est eigenvalue of the matrix Q. This approach has then been generalized and improved by many people (see e.g. Delorme and Poljak, 1993a,b; Poljak and Rendl, 1995; Billionnet and Elloumi, 2007) leading to bounds equivalent to the ones obtained by a semi- definite formulation presented in Goemans and Williamson (1995). More recently further improvements over the latter have been introduced, e.g. in Malik et al. (2006) and Ben-Ameur and Neto (2008a). Let us introduce some notation. The eigenvalues of the matrix Q will be noted k 1 ðQ Þ 6 k 2 ðQ Þ 6 ... 6 k n ðQ Þ (or more simply k 1 6 k 2 6 ... 6 k n when clear from the context) and corresponding unit (in Euclidean norm) and pairwise orthogonal eigenvectors: v 1 ; ... ; v n . The jth entry of vector v i is noted v ij . Given some set of vec- tors a 1 ; ... ; a q 2 R n ; q 2 N, we note Linða 1 ; ... ; a q Þ the subspace spanned by these vectors. Given some vector y 2 f1; 1g n ; distðy; Linðv 1 ; ... ; v p ÞÞ stands for the Euclidean distance between the vector y and Linðv 1 ; ... ; v p Þ, i.e. distðy; Linðv 1 ; ... ; v p ÞÞ ¼ ky  y p k 2 where y p stands for the orthogonal projection of y onto Linðv 1 ; ... ; v p Þ and kk 2 represents the Euclidean norm. Given some index j 2f1; ... ; ng; d j will denote the distance between the set f1; 1g n and the subspace that is spanned by the eigenvectors v 1 ; ... ; v j , i.e. minfdistðy; Linðv 1 ; ... ; v j ÞÞjy 2 f1; 1g n g. Notice that d j depends on a particular spectral decomposition of the matrix Q when there is an eigenvalue with multiplicity greater than 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.02.042 * Corresponding author. Tel.: +33 1 60764466. E-mail addresses: Walid.Benameur@it-sudparis.eu (W. Ben-Ameur), Jose.Ne to@it-sudparis.eu (J. Neto). European Journal of Operational Research 207 (2010) 15–24 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor