American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming
  • JIMO Home
  • This Issue
  • Next Article
    Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces
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

Yong Xia 1,

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.
Keywords: semidefinite programming., Orthogonal similarity, quadratic assignment problem, convex hull, lower bound.
Mathematics Subject Classification: Primary: 90C27; Secondary: 90C2.
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), 41-55. 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), 254-265. 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), 341-357. 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), 24-42.  Google Scholar

[5]

R. E. Burkard, Locations with spatial interactions: The quadratic assignment problem, in "Discrete Location Theory" (eds. P. B.Mirchandani and R. L.Francis), Wiley, New York, (1991), 387-437.  Google Scholar

[6]

R. E. Burkard, E. Çela, P. M. Pardalos and L. S. Pitsoulis, The quadratic assignment Problem, in "Handbook of Combinatorial Optimization" (eds. D.-Z. Du and P. M. Pardalos), 3, Kluwer, (1998), 241-337.  Google Scholar

[7]

E. Çela, "The Quadratic Assignment Problem: Theory and Algorithms," Kluwer Academic Publishers, Dordrecht, 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), 1008-1022. 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), 55-86. Google Scholar

[10]

C. S. Edwards, A branch and bound algorithm for the Koopmans-Beckmann quadratic assignment problem, Mathematical Programming Study, 13 (1980), 35-52.  Google Scholar

[11]

P. A. Fillmore and J. P. Williams, Some convexity theorems for matrices, Glasgow Math. J., 12 (1971), 110-117. doi: 10.1017/S0017089500001221.  Google Scholar

[12]

G. Finke, R. E. Burkard and F. Rendl, Quadratic assignment problems, Ann. Discrete Math., 31 (1987), 61-82. doi: 10.1016/S0304-0208(08)73232-8.  Google Scholar

[13]

M. Grant and S. Boyd, "{CVX}: {Matlab} Software for Disciplined Convex Programming," http://cvxr.com/cvx. version 1. 21 (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), 727-739. doi: 10.1287/moor.17.3.727.  Google Scholar

[15]

G. G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge University Press, London and New York, 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), 657-690. 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, New York, NY, 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), 41-45. 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 "Quadratic Assignment and Related Problems" (eds. P. M. Pardalos and H. Wolkowicz), 16 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science,AMS. 16 (1994), 1-42.  Google Scholar

[20]

S. Sahni and T. Gonzalez, P-complete approximation problems, Journal of the Association of Computing Machinery, 23 (1976), 555-565. doi: 10.1145/321958.321975.  Google Scholar

[21]

Y. Xia, Second order cone programming relaxation for quadratic assignment problems, Optimization Methods $&$ Software, 23 (2008), 441-449. 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), 881-892. 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), 769-778. doi: 10.1142/S0217595909002468.  Google Scholar

show all references

References:
[1]

K. M. Anstreicher and H. Wolkowicz, On lagrangian relaxation of quadratic matrix constraints, SIAM J. Matrix Anal. Appl., 22 (2000), 41-55. 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), 254-265. 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), 341-357. 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), 24-42.  Google Scholar

[5]

R. E. Burkard, Locations with spatial interactions: The quadratic assignment problem, in "Discrete Location Theory" (eds. P. B.Mirchandani and R. L.Francis), Wiley, New York, (1991), 387-437.  Google Scholar

[6]

R. E. Burkard, E. Çela, P. M. Pardalos and L. S. Pitsoulis, The quadratic assignment Problem, in "Handbook of Combinatorial Optimization" (eds. D.-Z. Du and P. M. Pardalos), 3, Kluwer, (1998), 241-337.  Google Scholar

[7]

E. Çela, "The Quadratic Assignment Problem: Theory and Algorithms," Kluwer Academic Publishers, Dordrecht, 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), 1008-1022. 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), 55-86. Google Scholar

[10]

C. S. Edwards, A branch and bound algorithm for the Koopmans-Beckmann quadratic assignment problem, Mathematical Programming Study, 13 (1980), 35-52.  Google Scholar

[11]

P. A. Fillmore and J. P. Williams, Some convexity theorems for matrices, Glasgow Math. J., 12 (1971), 110-117. doi: 10.1017/S0017089500001221.  Google Scholar

[12]

G. Finke, R. E. Burkard and F. Rendl, Quadratic assignment problems, Ann. Discrete Math., 31 (1987), 61-82. doi: 10.1016/S0304-0208(08)73232-8.  Google Scholar

[13]

M. Grant and S. Boyd, "{CVX}: {Matlab} Software for Disciplined Convex Programming," http://cvxr.com/cvx. version 1. 21 (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), 727-739. doi: 10.1287/moor.17.3.727.  Google Scholar

[15]

G. G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge University Press, London and New York, 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), 657-690. 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, New York, NY, 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), 41-45. 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 "Quadratic Assignment and Related Problems" (eds. P. M. Pardalos and H. Wolkowicz), 16 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science,AMS. 16 (1994), 1-42.  Google Scholar

[20]

S. Sahni and T. Gonzalez, P-complete approximation problems, Journal of the Association of Computing Machinery, 23 (1976), 555-565. doi: 10.1145/321958.321975.  Google Scholar

[21]

Y. Xia, Second order cone programming relaxation for quadratic assignment problems, Optimization Methods $&$ Software, 23 (2008), 441-449. 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), 881-892. 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), 769-778. doi: 10.1142/S0217595909002468.  Google Scholar

[1]

Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial & Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177
[2]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, 2021, 15 (2) : 315-338. doi: 10.3934/ipi.2020070
[3]

Qinghong Zhang, Gang Chen, Ting Zhang. Duality formulations in semidefinite programming. Journal of Industrial & Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881
[4]

Kien Ming Ng, Trung Hieu Tran. A parallel water flow algorithm with local search for solving the quadratic assignment problem. Journal of Industrial & Management Optimization, 2019, 15 (1) : 235-259. doi: 10.3934/jimo.2018041
[5]

R. N. Gasimov, O. Ustun. Solving the quadratic assignment problem using F-MSG algorithm. Journal of Industrial & Management Optimization, 2007, 3 (2) : 173-191. doi: 10.3934/jimo.2007.3.173
[6]

Xiaojin Zheng, Zhongyi Jiang. Tighter quadratically constrained convex reformulations for semi-continuous quadratic programming. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2331-2343. doi: 10.3934/jimo.2020071
[7]

Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103
[8]

Chenchen Wu, Dachuan Xu, Xin-Yuan Zhao. An improved approximation algorithm for the $2$-catalog segmentation problem using semidefinite programming relaxation. Journal of Industrial & Management Optimization, 2012, 8 (1) : 117-126. doi: 10.3934/jimo.2012.8.117
[9]

Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507
[10]

Zhi-Bin Deng, Ye Tian, Cheng Lu, Wen-Xun Xing. Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming. Journal of Industrial & Management Optimization, 2018, 14 (2) : 625-636. doi: 10.3934/jimo.2017064
[11]

Jiani Wang, Liwei Zhang. Statistical inference of semidefinite programming with multiple parameters. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1527-1538. doi: 10.3934/jimo.2019015
[12]

Daniel Heinlein, Ferdinand Ihringer. New and updated semidefinite programming bounds for subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 613-630. doi: 10.3934/amc.2020034
[13]

Shouhong Yang. Semidefinite programming via image space analysis. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1187-1197. doi: 10.3934/jimo.2016.12.1187
[14]

Christine Bachoc, Alberto Passuello, Frank Vallentin. Bounds for projective codes from semidefinite programming. Advances in Mathematics of Communications, 2013, 7 (2) : 127-145. doi: 10.3934/amc.2013.7.127
[15]

Bettina Klaus, Frédéric Payot. Paths to stability in the assignment problem. Journal of Dynamics & Games, 2015, 2 (3&4) : 257-287. doi: 10.3934/jdg.2015004
[16]

Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193
[17]

Wan Nor Ashikin Wan Ahmad Fatthi, Adibah Shuib, Rosma Mohd Dom. A mixed integer programming model for solving real-time truck-to-door assignment and scheduling problem at cross docking warehouse. Journal of Industrial & Management Optimization, 2016, 12 (2) : 431-447. doi: 10.3934/jimo.2016.12.431
[18]

Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111
[19]

Srimanta Bhattacharya, Sushmita Ruj, Bimal Roy. Combinatorial batch codes: A lower bound and optimal constructions. Advances in Mathematics of Communications, 2012, 6 (2) : 165-174. doi: 10.3934/amc.2012.6.165
[20]

Cristian Dobre. Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 367-378. doi: 10.3934/naco.2013.3.367

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (153)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy