Abstract
Quadratic assignment problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semi-definite relaxation models for QAPs based on matrix splitting has been proposed (Mittelmann and Peng, SIAM J Optim 20:3408–3426, 2010; Peng et al., Math Program Comput 2:59–77, 2010). In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the resulting relaxation model is easy to solve and able to provide a strong bound. For this, we first introduce a new notion of the so-called redundant and non-redundant matrix splitting and show that the relaxation based on a non-redundant matrix splitting can provide a stronger bound than a redundant one. Then we propose to follow the minimal trace principle to find a non-redundant matrix splitting via solving an auxiliary semi-definite programming problem. We show that applying the minimal trace principle directly leads to the so-called orthogonal matrix splitting introduced in (Peng et al., Math Program Comput 2:59–77, 2010). To find other non-redundant matrix splitting schemes whose resulting relaxation models are relatively easy to solve, we elaborate on two splitting schemes based on the so-called one-matrix and the sum-matrix. We analyze the solutions from the auxiliary problems for these two cases and characterize when they can provide a non-redundant matrix splitting. The lower bounds from these two splitting schemes are compared theoretically. Promising numerical results on some large QAP instances are reported, which further validate our theoretical conclusions.
Similar content being viewed by others
Notes
The minimal trace principle is chosen because, as shown in our analysis in Sect. 2, the final solution matrix following the minimal principle has the minimal rank and the rank information on the splitting matrix can be further used to reduce the memory requirement and simplify the relaxation model as discussed in Sect. 4.
For simplicity of discussion, in all the theoretical analysis of this work, we consider only the basic model 2 which is slightly different from the full SDR model to be described in Sect. 4. However, since in the full model we only add some convex constraints on the elements of \(Y\), one can easily extend the results for the basic model to the full model.
We note that a similar approach (called the reduction method) has been used to improve the GLB and eigenvalue bound for QAPs with nonsymmetric matrices in the literature [7, 8, 12, 29]. One simple choice is \(u=\min (B_\mathrm{off})\). For more details on the reduction method, we refer to Sect. 7.5.2 of [7].
References
Adams, W.P., Johnson, T.A.: Improved linear programming-based lower bounds for the quadratic assignment problem. In: Pardalos, P.M., Wolkowicz, H. (eds.) Quadratic Assignment and Related Problems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16, pp. 43–75. AMS, Rhode Island (1994)
Adams, W., Guignard, M., Hahn, P., Hightower, W.: A level-2 reformulation-linearization technique bound for the quadratic assignment problem. Eur. J. Oper. Res. 180, 983–996 (2007)
Anstreicher, K., Brixius, N.: A new lower bound via projection for the quadratic assignment problem. Math. Program. 80, 341–357 (2001)
Ben-David, G., Malah, D.: Bounds on the performance of vector-quantizers under channel errors. IEEE Trans. Inf. Theory 51, 2227–2235 (2005)
Burer, S., Vandenbussche, D.: Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16, 726–750 (2006)
Burkard, R., Karisch, S., Rendl, F.: QAPLIB—a quadratic assignment problem library. J. Glob. Optim. 10, 391–403 (1997). Recent updates on QAPLIB are avaliable at http://www.seas.upenn.edu/qaplib/
Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Conrad, K.: Das quadratische Zuweisungsproblem und zwei seiner Spezialfalle. Mohr Siebeck, Tubingen (1971)
de Carvalho, S.A., Rahmann, S.: Microarray layout as a quadratic assignment problem. Proc. Ger. Conf. Bioinform. 83, 11–20 (2006)
De Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Program. 122, 225–246 (2010)
Ding, Y., Wolkowicz, H.: A low-dimensional semidefinite relaxation for the quadratic assignment problem. Math. Oper. Res. 34, 1008–1022 (2009)
Edwards, C.: A branch and bound algorithm for the Koopmans–Beckmann quadratic assignment problem. Comb. Optim. II, 35–52 (1980)
Gilmore, P.: Optimal and suboptimal algorithms for the quadratic assignment problem. SIAM J. Appl. Math. 10, 305–313 (1962)
Grant, M., Boyd, S., Ye, Y.: CVX: Matlab software for disciplined convex programming. http://www.stanford.edu/boyd/cvx. Accessed 2013
Hadley, S.W., Rendl, F., Wolkowicz, H.: A new lower bound via projection for the quadratic assignment problem. Math. Oper. Res. 17, 727–739 (1992)
Hahn, P., Grant, T.: Lower bounds for the quadratic assignment problem based upon a dual formulation. Oper. Res. 46, 912–922 (1998)
Hahn, P., Anjos, M., Burkard, R.E., Karisch, S.E., Rendl, F.: QAPLIB—a quadratic assignment problem library. http://www.seas.upenn.edu/qaplib/. Accessed 2013
Hahn, P.M., Zhu, Y.R., Guignard, M., Hightower, W.L., Saltzman, M.J.: A level-3 reformulation-linearization technique-based bound for the quadratic assignment problem. INFORMS J. Comput. 24, 202–209 (2012)
Hanan, M., Kurtzberg, J.: Placement techniques. Des. Autom. Digital Syst. 1, 213–282 (1972)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Jansson, C., Chaykin, D., Keil, C.: Rigorous error bounds for the optimal value in semidefinite programming. SIAM J. Numer. Anal. 46, 180–200 (2007)
Koopmans, T., Beckmann, M.: Assignment problems and the location of economic activities. Econometrica 25, 53–76 (1957)
Lawler, E.: The quadratic assignment problem. Manage. Sci. 9, 589–599 (1963)
Loiola, E., Abreu, N., Boaventura-Netto, P., Hahn, P., Querido, T.: A survey for the quadratic assignment problem. Eur. J. Oper. Res. 176, 657–690 (2007)
Mittelmann, H., Peng, J.: Estimating bounds for quadratic assignment problems associated with Hamming and Manhattan distance matrices based on semidefinite programming. SIAM J. Optim. 20, 3408–3426 (2010)
Mukherjee, L., Singh, V., Peng, J., Xu, J., Zeitz, M., Berezney, R.: Generalized median graphs and applications. J Combin. Optim. 17, 21–44 (2009)
Peng, J., Mittelmann, H., Li, X.: A new relaxation framework for quadratic assignment problems based on matrix splitting. Math. Program. Comput. 2, 59–77 (2010)
Rendl, F., Sotirov, R.: Bounds for the quadratic assignment problem using the bundle method. Math. Program. 109, 505–524 (2007)
Roucairol, C.: A reduction method for quadratic assignment problems. Methods Oper. Res. 32, 185–187 (1979)
Taillard, E.: Comparison of iterative searches for the quadratic assignment problem. Locat. Sci. 3, 87–105 (1995)
Theobald, C.M.: An inequality for the trace of the product of two symmetric matrices. Math. Proc. Camb. Philos. Soc. 77, 77–265 (1975)
Toh, K., Todd, M., Tutuncu, R.: SDPT3: Matlab software package for semidefinite programming. Optim. Methods Softw. 11, 545–581 (1999)
Zhao, Q., Karisch, S., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Combin. Optim. 2, 71–109 (1998)
Zhao, X., Sun, D., Toh, K.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)
Acknowledgments
We would like to thank the two anonymous referees and the Associate Editor for their helpful suggestions that led to substantial improvement on the presentation of the paper. We also thank Etienne de Klerk for pointing out some missing constraints in our previous implementation of the SDRMS-SUM model in CVX. This work was jointly supported by AFOSR Grant FA9550-09-1-0098 and NSF Grant DMS 09-15240 ARRA, the National Natural Science Foundation of China under Grants 11071219 and 11371324, and the Zhejiang Provincial Natural Science Foundation of China under Grant LY13A010012.
Author information
Authors and Affiliations
Corresponding author
Appendix: Proofs of Theorems 2 and 3
Appendix: Proofs of Theorems 2 and 3
Proof of Theorem 2
Denote the optimal solution to the MTMS-PSD problem by \((B^*_1, B^*_2)\). We first show \((B^*_1, B^*_2)=(B^+, B^-)\). Let \(P\) be the projection matrix defined by
It follows immediately that
where the first inequality follows from the relation
Here \(I\) denotes the identity matrix in \(\mathfrak {R}^{n\times n}\). Similarly, one has
Therefore, we have
and the equality holds if and only if
Since all the matrices \(B_1^*, B_2^*, P\) and \( I-P\) are positive semi-definite. Relation holds if and only if
Since \(B_1^*-B_2^* = B = B^+ - B^-\), we thus have
It remains to show that the matrix splitting \((B^+, B^-)\) is non-redundant. Suppose to the contrary that \((B^+, B^-)\) is a redundant splitting of \(B\), i.e., there exists \(R\ne 0 \succeq 0\) such that
Then we have
which contradicts to the relation \(\mathrm{Tr}(B^+B^-)=0\). This finishes the proof of the theorem.\(\square \)
We next cite a well-known result for matrix product from [31] that will be used in the proof of Theorem 3.
Lemma 3
Let \(A,B\in {\textsc {S}}^n\) with the eigenvalues \(\lambda _i(A)\) and \(\lambda _i(B)\), \(i=1,\ldots ,n\) listed in nonincreasing order. Then
where the equality holds if and only if there is an orthogonal matrix \(P\) whose columns form a common set of eigenvectors for A and B and are ordered with respect to \(\{\lambda _i(A)\}_{i=1}^n\) and \(\{\lambda _i(B)\}_{i=1}^n\), such that \(P^{-1}AP\) and \(P^{-1}BP\) are diagonal.
Proof of Theorem 3
Since \((\alpha ,\beta )\) is an optimal solution of problem (23), there exists \(U\in {\textsc {S}}^n\) such that
From (42) to (44), we obtain directly
From (44) and (45), we follow that
This implies that \(U\) and \(\alpha E +\beta I - B\) can commute. By Theorem 1.3.12 in [20], \(U\) and \(\alpha E +\beta I - B\) are simultaneously diagonalizable. Since \(U\in {\textsc {S}}^n\), \(\alpha E +\beta I - B\in {\textsc {S}}^n\), there is an orthogonal matrix \(P\) such that \(P^{-1}UP\) and \(P^{-1}(\alpha E +\beta I - B)P\) are diagonal. So, we have
which in turn by (45) implies that
Due to the minimal trace principle, we have \(m=\mathrm{Rank}(\alpha E+\beta I -B) < n\). Since \(\alpha E +\beta I - B\succeq 0\), we assume that \(\lambda _i(\alpha E +\beta I - B)>0\), \(i=1,\ldots ,m\). The above equality (48) then yields \(\lambda _i(U)=0\), \(i=1,\ldots ,m\).
We now prove \(UE\ne EU\). Suppose to the contrary that \(UE= EU\). By Theorem 1.3.12 in [20], \(U\) and \(E\) are simultaneously diagonalizeable. Let
Note that the eigenvalues of \(E\) are \(0,\ldots ,0,n\). Therefore, we have
which by (42) implies \(\lambda _n(U)=1\). Hence, we infer from (49) that
this contradicts (43).
Because \(UE\ne EU\), from (47) we obtain \(UB\ne BU\). Since \(U\in {\textsc {S}}^n\) and \(B\in {\textsc {S}}^n\), by Theorem 1.3.12 in [20], \(U\) and \(B\) are not simultaneously diagonalizable. Now using Lemma 3, we have
which, together with (46), yields
Let \(\lambda _{\max }(B)\) be the largest eigenvalue of \(B\). Note that \(\sum _{i=1}^n\lambda _i(U)=\mathrm{Tr}(U)=n\). Also, \(\lambda _i(U)\ge 0\) for all \(i\) since \(U\succeq 0\). It then follows from (50) that
On the other hand, from (45), we have
This means that
If \(\alpha =0\), then the combination of (51) and (52) leads to a contradiction.
If \(\alpha <0\). Let \(\rho (B)\) be the spectral radius of \(B\). Since \(B\in {\textsc {S}}^n\) is non-negative, by Theorem 8.3.1 in [20], then \(\rho (B)\) is an eigenvalue of \(B\) and there exists nontrivial \(\hat{x} \ge 0\in {\mathfrak {R}}^n\) such that \(B\hat{x}=\rho (B)\hat{x}\). Without loss of generality, we can further assume that \(\Vert \hat{x}\Vert _2=1\). Thus we have \(\hat{x}^TB\hat{x}=\rho (B)\). Since \(\hat{x}\ge 0\), it holds \(\hat{x}^T(E-I)\hat{x}\ge 0\). It follows from (52) that
which contradicts to (51). Therefore, we can conclude \(\alpha >0\). This finishes the proof of the theorem.\(\square \)
Rights and permissions
About this article
Cite this article
Peng, J., Zhu, T., Luo, H. et al. Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting. Comput Optim Appl 60, 171–198 (2015). https://doi.org/10.1007/s10589-014-9663-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-014-9663-y