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Extended canonical duality and conic programming for solving 01 quadratic programming problems
1.  Department of Mathematical Sciences, Tsinghua University, Beijing, China 
2.  Department of Industrial and System Engineering, North Carolina State University, Raleigh, NC 27695, United States 
References:
[1] 
K. Allemand, K. Fukuda, T. M. Liebling and E. Steiner, A polynomial case of unconstrained zeroone quadratic optimization, Mathematical Programming, 91 (2001), 4952. 
[2] 
S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479495. doi: 10.1007/s101070080223z. 
[3] 
S.C. Fang, D. Y. Gao, R.L. Sheu and S.Y. Wu, Canonical dual approach to solving 01 quadratic programming problem, J. Industrial and Management Optimization, 3 (2007), 125142. 
[4] 
S.C. Fang, D. Y. Gao, R.L. Sheu and W. Xing, Global optimization for a class of fractional programming problems, J. Global Optimization, 45 (2009), 337353. doi: 10.1007/s1089800893787. 
[5] 
D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optimization, 17 (2000), 127160. doi: 10.1023/A:1026537630859. 
[6] 
D. Y. Gao, Advances in canonical duality theory with applications to global optimization, available at: http://www.math.vt.edu/people/gao/papers/focapo08.pdf. 
[7] 
M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman, San Francisco, CA, 1979. 
[8] 
M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), 11151145. doi: 10.1145/227683.227684. 
[9] 
V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Nonconvex quadratic minimization problems with quadratic constraints: Global optimality conditions, Mathematical Programming, 110 (2007), 521541. doi: 10.1007/s1010700600125. 
[10] 
Q. Jin, S.C. Fang and W. Xing, On the global optimality of generalized trust region subproblems, Optimization. doi: DOI:10.1080/02331930902995236. 
[11] 
E. Klerk and D. V. Pasechnik, Approximation of the stability number of a graph via copositive programming, SIAM J. Optimization, 12 (2002), 875892. doi: 10.1137/S1052623401383248. 
[12] 
C. Lu, Z. Wang and W. Xing, An improved lower bound and approximation algorithm for binary constrained quadratic programming problem, J. Global Optimization. doi: DOI: 10.1007/s1089800995041. 
[13] 
A. BenIsrael and T. N. E. Greville, "Generalized Inverses," SpringerVerlag, 2003. 
[14] 
P. M. Pardalos and G. P. Rodgers, Computational aspects of a branch and bound algorithm for quadratic zeroone programming, Computing, 45 (1990), 131144. doi: 10.1007/BF02247879. 
[15] 
J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246267. doi: 10.1287/moor.28.2.246.14485. 
[16] 
X. Sun, C. Liu, D. Li and J. Gao, On duality gap in binary quadratic programming, aviable at: http://www.optimizationonline.org/DB_FILE/2010/01/2512.pdf. 
[17] 
Z. Wang, S.C. Fang, D. Y. Gao and W. Xing, Global extremal conditions for multiinteger quadratic programming, J. Industrial and Management Optimization, 4 (2008), 213225. 
[18] 
Z. Wang, S.C. Fang, D. Y. Gao and W. Xing, Canonical dual approach to solving the maximum cut problem, Working Paper, to appear in J. Global Optimization. 
[19] 
L. A. Wolsey, "Integer Programming," WileyInterscience, 1998. 
[20] 
W. Xing, S.C. Fang, D. Y. Gao and L. Zhang, Canonical duality solutions to quadratic programming over a quadratic constraint, Proceedings of ICOTA7, (2007), 3536. 
show all references
References:
[1] 
K. Allemand, K. Fukuda, T. M. Liebling and E. Steiner, A polynomial case of unconstrained zeroone quadratic optimization, Mathematical Programming, 91 (2001), 4952. 
[2] 
S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479495. doi: 10.1007/s101070080223z. 
[3] 
S.C. Fang, D. Y. Gao, R.L. Sheu and S.Y. Wu, Canonical dual approach to solving 01 quadratic programming problem, J. Industrial and Management Optimization, 3 (2007), 125142. 
[4] 
S.C. Fang, D. Y. Gao, R.L. Sheu and W. Xing, Global optimization for a class of fractional programming problems, J. Global Optimization, 45 (2009), 337353. doi: 10.1007/s1089800893787. 
[5] 
D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optimization, 17 (2000), 127160. doi: 10.1023/A:1026537630859. 
[6] 
D. Y. Gao, Advances in canonical duality theory with applications to global optimization, available at: http://www.math.vt.edu/people/gao/papers/focapo08.pdf. 
[7] 
M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman, San Francisco, CA, 1979. 
[8] 
M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), 11151145. doi: 10.1145/227683.227684. 
[9] 
V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Nonconvex quadratic minimization problems with quadratic constraints: Global optimality conditions, Mathematical Programming, 110 (2007), 521541. doi: 10.1007/s1010700600125. 
[10] 
Q. Jin, S.C. Fang and W. Xing, On the global optimality of generalized trust region subproblems, Optimization. doi: DOI:10.1080/02331930902995236. 
[11] 
E. Klerk and D. V. Pasechnik, Approximation of the stability number of a graph via copositive programming, SIAM J. Optimization, 12 (2002), 875892. doi: 10.1137/S1052623401383248. 
[12] 
C. Lu, Z. Wang and W. Xing, An improved lower bound and approximation algorithm for binary constrained quadratic programming problem, J. Global Optimization. doi: DOI: 10.1007/s1089800995041. 
[13] 
A. BenIsrael and T. N. E. Greville, "Generalized Inverses," SpringerVerlag, 2003. 
[14] 
P. M. Pardalos and G. P. Rodgers, Computational aspects of a branch and bound algorithm for quadratic zeroone programming, Computing, 45 (1990), 131144. doi: 10.1007/BF02247879. 
[15] 
J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246267. doi: 10.1287/moor.28.2.246.14485. 
[16] 
X. Sun, C. Liu, D. Li and J. Gao, On duality gap in binary quadratic programming, aviable at: http://www.optimizationonline.org/DB_FILE/2010/01/2512.pdf. 
[17] 
Z. Wang, S.C. Fang, D. Y. Gao and W. Xing, Global extremal conditions for multiinteger quadratic programming, J. Industrial and Management Optimization, 4 (2008), 213225. 
[18] 
Z. Wang, S.C. Fang, D. Y. Gao and W. Xing, Canonical dual approach to solving the maximum cut problem, Working Paper, to appear in J. Global Optimization. 
[19] 
L. A. Wolsey, "Integer Programming," WileyInterscience, 1998. 
[20] 
W. Xing, S.C. Fang, D. Y. Gao and L. Zhang, Canonical duality solutions to quadratic programming over a quadratic constraint, Proceedings of ICOTA7, (2007), 3536. 
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