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July  2013, 9(3): 689-701. doi: 10.3934/jimo.2013.9.689

## Convex hull of the orthogonal similarity set with applications in quadratic assignment problems

 1 State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing, 100191, China

Received  August 2012 Revised  November 2012 Published  April 2013

In this paper, we study thoroughly the convex hull of the orthogonal similarity set and give a new representation. When applied in quadratic assignment problems, it motivates two new lower bounds. The first is equivalent to the projected eigenvalue bound, while the second highly outperforms several well-known lower bounds in literature.
Citation: Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial & Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689
##### References:
 [1] K. M. Anstreicher and H. Wolkowicz, On lagrangian relaxation of quadratic matrix constraints,, SIAM J. Matrix Anal. Appl., 22 (2000), 41.  doi: 10.1137/S0895479898340299.  Google Scholar [2] K. M. Anstreicher, Eigenvalue bounds versus semidefinite relaxations for the quadratic assignment problem,, SIAM Journal on Optimization, 11 (2001), 254.  doi: 10.1137/S1052623499354904.  Google Scholar [3] K. M. Anstreicher and N. W. Brixius, A new bound for the quadratic assignment problem based on convex quadratic programming,, Mathematical Programming Ser. A, 89 (2001), 341.  doi: 10.1007/PL00011402.  Google Scholar [4] K. M. Anstreicher, Recent advances in the solution of quadratic assignment Problems,, Mathematical Programming Ser. B, 97 (2003), 24.   Google Scholar [5] R. E. Burkard, Locations with spatial interactions: The quadratic assignment problem,, in, (1991), 387.   Google Scholar [6] R. E. Burkard, E. Çela, P. M. Pardalos and L. S. Pitsoulis, The quadratic assignment Problem,, in, 3 (1998), 241.   Google Scholar [7] E. Çela, "The Quadratic Assignment Problem: Theory and Algorithms,", Kluwer Academic Publishers, (1998).   Google Scholar [8] Y. Ding and H. Wolkowicz, A low-dimensional semidefinite relaxation for the quadratic assignment problem,, Mathematics of Operations Research, 34 (2009), 1008.  doi: 10.1287/moor.1090.0419.  Google Scholar [9] C. S. Edwards, The derivation of a greedy approximator for the Koopmans-Beckmann quadratic assignment problem,, Proceedings of the 77-th Combinatorial Programming Conference (CP77) (1977), (1977), 55.   Google Scholar [10] C. S. Edwards, A branch and bound algorithm for the Koopmans-Beckmann quadratic assignment problem,, Mathematical Programming Study, 13 (1980), 35.   Google Scholar [11] P. A. Fillmore and J. P. Williams, Some convexity theorems for matrices,, Glasgow Math. J., 12 (1971), 110.  doi: 10.1017/S0017089500001221.  Google Scholar [12] G. Finke, R. E. Burkard and F. Rendl, Quadratic assignment problems,, Ann. Discrete Math., 31 (1987), 61.  doi: 10.1016/S0304-0208(08)73232-8.  Google Scholar [13] M. Grant and S. Boyd, "{CVX}: {Matlab} Software for Disciplined Convex Programming,", , (2010).   Google Scholar [14] S. W. Hadley, F. Rendl and H. Wolkowicz, A new lower bound via projection for the quadratic assignment problem,, Math. Oper. Res., 17 (1992), 727.  doi: 10.1287/moor.17.3.727.  Google Scholar [15] G. G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", Cambridge University Press, (1952).   Google Scholar [16] E. M. Loiola, N. M. M. Abreu, P. O. Boaventura-Netto, P. Hahn and T. Querido, A survey for the quadratic assignment problem,, European Journal of Operational Research, 176 (2007), 657.  doi: 10.1016/j.ejor.2005.09.032.  Google Scholar [17] A. W. Marshall and I. Olkin, "Inequalities: Theory of Majorization and its Application,", Academic Press, (1979).   Google Scholar [18] M. L. Overton and R. S. Womersley, On the sum of the largest eigenvalues of a symmetric matrix,, SIAM J. Matrix Anal. Appl., 13 (1992), 41.  doi: 10.1137/0613006.  Google Scholar [19] P. M. Pardalos, F. Rendl and H. Wolkowicz, The Quadratic Assignment Problem: A Survey and Recent Developments,, in, 16 (1994), 1.   Google Scholar [20] S. Sahni and T. Gonzalez, P-complete approximation problems,, Journal of the Association of Computing Machinery, 23 (1976), 555.  doi: 10.1145/321958.321975.  Google Scholar [21] Y. Xia, Second order cone programming relaxation for quadratic assignment problems,, Optimization Methods $&$ Software, 23 (2008), 441.  doi: 10.1080/10556780701843405.  Google Scholar [22] Y. Xia, New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problem,, Journal of Industrial and Management Optimization, 5 (2009), 881.  doi: 10.3934/jimo.2009.5.881.  Google Scholar [23] Y. Xia, Convex hull presentation of a quadratically constrained set and its application in solving quadratic programming problems,, Asia-Pacific Journal of Operational Research, 26 (2009), 769.  doi: 10.1142/S0217595909002468.  Google Scholar

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##### References:
 [1] K. M. Anstreicher and H. Wolkowicz, On lagrangian relaxation of quadratic matrix constraints,, SIAM J. Matrix Anal. Appl., 22 (2000), 41.  doi: 10.1137/S0895479898340299.  Google Scholar [2] K. M. Anstreicher, Eigenvalue bounds versus semidefinite relaxations for the quadratic assignment problem,, SIAM Journal on Optimization, 11 (2001), 254.  doi: 10.1137/S1052623499354904.  Google Scholar [3] K. M. Anstreicher and N. W. Brixius, A new bound for the quadratic assignment problem based on convex quadratic programming,, Mathematical Programming Ser. A, 89 (2001), 341.  doi: 10.1007/PL00011402.  Google Scholar [4] K. M. Anstreicher, Recent advances in the solution of quadratic assignment Problems,, Mathematical Programming Ser. B, 97 (2003), 24.   Google Scholar [5] R. E. Burkard, Locations with spatial interactions: The quadratic assignment problem,, in, (1991), 387.   Google Scholar [6] R. E. Burkard, E. Çela, P. M. Pardalos and L. S. Pitsoulis, The quadratic assignment Problem,, in, 3 (1998), 241.   Google Scholar [7] E. Çela, "The Quadratic Assignment Problem: Theory and Algorithms,", Kluwer Academic Publishers, (1998).   Google Scholar [8] Y. Ding and H. Wolkowicz, A low-dimensional semidefinite relaxation for the quadratic assignment problem,, Mathematics of Operations Research, 34 (2009), 1008.  doi: 10.1287/moor.1090.0419.  Google Scholar [9] C. S. Edwards, The derivation of a greedy approximator for the Koopmans-Beckmann quadratic assignment problem,, Proceedings of the 77-th Combinatorial Programming Conference (CP77) (1977), (1977), 55.   Google Scholar [10] C. S. Edwards, A branch and bound algorithm for the Koopmans-Beckmann quadratic assignment problem,, Mathematical Programming Study, 13 (1980), 35.   Google Scholar [11] P. A. Fillmore and J. P. Williams, Some convexity theorems for matrices,, Glasgow Math. J., 12 (1971), 110.  doi: 10.1017/S0017089500001221.  Google Scholar [12] G. Finke, R. E. Burkard and F. Rendl, Quadratic assignment problems,, Ann. Discrete Math., 31 (1987), 61.  doi: 10.1016/S0304-0208(08)73232-8.  Google Scholar [13] M. Grant and S. Boyd, "{CVX}: {Matlab} Software for Disciplined Convex Programming,", , (2010).   Google Scholar [14] S. W. Hadley, F. Rendl and H. Wolkowicz, A new lower bound via projection for the quadratic assignment problem,, Math. Oper. Res., 17 (1992), 727.  doi: 10.1287/moor.17.3.727.  Google Scholar [15] G. G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", Cambridge University Press, (1952).   Google Scholar [16] E. M. Loiola, N. M. M. Abreu, P. O. Boaventura-Netto, P. Hahn and T. Querido, A survey for the quadratic assignment problem,, European Journal of Operational Research, 176 (2007), 657.  doi: 10.1016/j.ejor.2005.09.032.  Google Scholar [17] A. W. Marshall and I. Olkin, "Inequalities: Theory of Majorization and its Application,", Academic Press, (1979).   Google Scholar [18] M. L. Overton and R. S. Womersley, On the sum of the largest eigenvalues of a symmetric matrix,, SIAM J. Matrix Anal. Appl., 13 (1992), 41.  doi: 10.1137/0613006.  Google Scholar [19] P. M. Pardalos, F. Rendl and H. Wolkowicz, The Quadratic Assignment Problem: A Survey and Recent Developments,, in, 16 (1994), 1.   Google Scholar [20] S. Sahni and T. Gonzalez, P-complete approximation problems,, Journal of the Association of Computing Machinery, 23 (1976), 555.  doi: 10.1145/321958.321975.  Google Scholar [21] Y. Xia, Second order cone programming relaxation for quadratic assignment problems,, Optimization Methods $&$ Software, 23 (2008), 441.  doi: 10.1080/10556780701843405.  Google Scholar [22] Y. Xia, New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problem,, Journal of Industrial and Management Optimization, 5 (2009), 881.  doi: 10.3934/jimo.2009.5.881.  Google Scholar [23] Y. Xia, Convex hull presentation of a quadratically constrained set and its application in solving quadratic programming problems,, Asia-Pacific Journal of Operational Research, 26 (2009), 769.  doi: 10.1142/S0217595909002468.  Google Scholar
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