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Global convergence of an inexact operator splitting method for monotone variational inequalities
Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems
1.  Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27606, United States 
2.  Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 
References:
[1] 
K. Allemand, K. Fukuda, T. M. Liebling and E. Steiner, A polynomial case of unconstrained zeroone quadratic optimization, Math. Program, 91 (2001), 4952. 
[2] 
A. BenIsrael and T. N. E. Greville, "Generalized Inverses: Theory and Applications," 2nd edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 15, SpringerVerlag, New York, 2003. 
[3] 
A. Billionnet and F. Calmels, Linear programming for the 01 quadratic knapsack problem, European Journal of Operational Research, 92 (1996), 310325. doi: 10.1016/03772217(94)002290. 
[4] 
A. Billionnet, A. Faye and E. Soutif, A new upper bound for the 01 quadratic knapsack problem, European Journal of Operational Research, 113 (1999), 664672. doi: 10.1016/S03772217(97)004141. 
[5] 
D. Bienstock, Computational study of a family of mixedinteger quadratic programming problems, Math. Program, 74 (1996), 121140. doi: 10.1007/BF02592208. 
[6] 
I. M. Bomze, Global optimization: A quadratic programming perspective, in "Nonlinear Optimization," Lecture Notes in Mathematics, 1989, Springer, Berlin, (2010), 153. 
[7] 
I. M. Bomze and F. Jarre, A note on Burer's copositive representation of mixedbinary QPs, Optimization Letter, 4 (2010), 465472. doi: 10.1007/s1159001001741. 
[8] 
S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Math. Program., 120 (2009), 479495. doi: 10.1007/s101070080223z. 
[9] 
S. Bundfuss and M. Dür, "An Adaptive Linear Approximation Algorithm for Copositive Programs," Manuscript, Department of Mathematics, Technische Universitat Darmstadt, Darmstadt, Germany, 2008. 
[10] 
S.C. Fang, D. Y. Gao, R.L. Sheu and S.Y. Wu, Canonical dual approach to solving 01 quadratic programming problems, Journal of Industrial and Management Optimization, 4 (2008), 125142. 
[11] 
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. 
[12] 
D. Y. Gao, Advances in canonical duality theory with applications to global optimization,, Available from: \url{http://www.math.vt.edu/people/gao/papers/focapo08.pdf}., (). 
[13] 
M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NPCompleteness," A Series of Books in the Mathematical Sciences, W. H. Freeman and Co., San Francisco, CA, 1979. 
[14] 
G. T. Herman, "Image Reconstruction from Projections: The Fundamentals of Computerized Tomography," Computer Science and Applied Mathematics. Academic Press,Inc., New YorkLondon, 1980. 
[15] 
V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Nonconvex quadratic minimization problems with quadratic constraints: Global optimality conditions, Math. Program., 110 (2007), 521541. doi: 10.1007/s1010700600125. 
[16] 
E. de Klerk and D. V. Pasechnik, Approximation of the stability number of a graph via copositive programming, SIAM J. Optim., 12 (2002), 875892. doi: 10.1137/S1052623401383248. 
[17] 
C. Lu, S.C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, working paper, 2010. 
[18] 
C. Lu, Z. Wang, W. Xing and S.C. Fang, Extended canonical duality and conic programming for solving 01 quadratic programming problems, Journal of Industrial and Management Optimization, 6 (2010), 779793. doi: 10.3934/jimo.2010.6.779. 
[19] 
C. Lemaréchal and F. Oustry, SDP relaxations in combinatorial optimization from a Lagrangian viewpoint, in "Advances in Convex Analysis and Global Optimization" (eds. N. Hadijsavvas and P. M. Paradalos), Nonconvex Optim. Appl., 54, Kluwer Acad. Publ., Dordrecht, (2001), 119134. 
[20] 
J. B. Lasserre, Global optimization with polynomials and the problem of moments,, SIAM J. Optimization, 11 (): 796. doi: 10.1137/S1052623400366802. 
[21] 
P. Lötstedt, Solving the minimal least squares problem subject to bounds on the variables, BIT, 24 (1984), 206224. 
[22] 
P. Parrilo, "Structured Semidefinite Programs and SemiAlgebraic Geometry Methods in Robustness and Optimization," Ph.D. Thesis, California Institute of Technology, 2000. 
[23] 
J. F. Strum and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246267. doi: 10.1287/moor.28.2.246.14485. 
[24] 
X. Sun, C. Liu, D. Li and J. Gao, On duality gap in binary quadratic programming,, Available from: \url{http://www.optimizationonline.org/DB_FILE/2010/01/2512.pdf}., (). 
[25] 
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. doi: 10.3934/jimo.2008.4.213. 
[26] 
L. F. Zuluage, J. Vera and J. Peña, LMI approximations for cones of positive semidefinite forms, SIAM J. Optimization, 16 (2006), 10761091. 
show all references
References:
[1] 
K. Allemand, K. Fukuda, T. M. Liebling and E. Steiner, A polynomial case of unconstrained zeroone quadratic optimization, Math. Program, 91 (2001), 4952. 
[2] 
A. BenIsrael and T. N. E. Greville, "Generalized Inverses: Theory and Applications," 2nd edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 15, SpringerVerlag, New York, 2003. 
[3] 
A. Billionnet and F. Calmels, Linear programming for the 01 quadratic knapsack problem, European Journal of Operational Research, 92 (1996), 310325. doi: 10.1016/03772217(94)002290. 
[4] 
A. Billionnet, A. Faye and E. Soutif, A new upper bound for the 01 quadratic knapsack problem, European Journal of Operational Research, 113 (1999), 664672. doi: 10.1016/S03772217(97)004141. 
[5] 
D. Bienstock, Computational study of a family of mixedinteger quadratic programming problems, Math. Program, 74 (1996), 121140. doi: 10.1007/BF02592208. 
[6] 
I. M. Bomze, Global optimization: A quadratic programming perspective, in "Nonlinear Optimization," Lecture Notes in Mathematics, 1989, Springer, Berlin, (2010), 153. 
[7] 
I. M. Bomze and F. Jarre, A note on Burer's copositive representation of mixedbinary QPs, Optimization Letter, 4 (2010), 465472. doi: 10.1007/s1159001001741. 
[8] 
S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Math. Program., 120 (2009), 479495. doi: 10.1007/s101070080223z. 
[9] 
S. Bundfuss and M. Dür, "An Adaptive Linear Approximation Algorithm for Copositive Programs," Manuscript, Department of Mathematics, Technische Universitat Darmstadt, Darmstadt, Germany, 2008. 
[10] 
S.C. Fang, D. Y. Gao, R.L. Sheu and S.Y. Wu, Canonical dual approach to solving 01 quadratic programming problems, Journal of Industrial and Management Optimization, 4 (2008), 125142. 
[11] 
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. 
[12] 
D. Y. Gao, Advances in canonical duality theory with applications to global optimization,, Available from: \url{http://www.math.vt.edu/people/gao/papers/focapo08.pdf}., (). 
[13] 
M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NPCompleteness," A Series of Books in the Mathematical Sciences, W. H. Freeman and Co., San Francisco, CA, 1979. 
[14] 
G. T. Herman, "Image Reconstruction from Projections: The Fundamentals of Computerized Tomography," Computer Science and Applied Mathematics. Academic Press,Inc., New YorkLondon, 1980. 
[15] 
V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Nonconvex quadratic minimization problems with quadratic constraints: Global optimality conditions, Math. Program., 110 (2007), 521541. doi: 10.1007/s1010700600125. 
[16] 
E. de Klerk and D. V. Pasechnik, Approximation of the stability number of a graph via copositive programming, SIAM J. Optim., 12 (2002), 875892. doi: 10.1137/S1052623401383248. 
[17] 
C. Lu, S.C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, working paper, 2010. 
[18] 
C. Lu, Z. Wang, W. Xing and S.C. Fang, Extended canonical duality and conic programming for solving 01 quadratic programming problems, Journal of Industrial and Management Optimization, 6 (2010), 779793. doi: 10.3934/jimo.2010.6.779. 
[19] 
C. Lemaréchal and F. Oustry, SDP relaxations in combinatorial optimization from a Lagrangian viewpoint, in "Advances in Convex Analysis and Global Optimization" (eds. N. Hadijsavvas and P. M. Paradalos), Nonconvex Optim. Appl., 54, Kluwer Acad. Publ., Dordrecht, (2001), 119134. 
[20] 
J. B. Lasserre, Global optimization with polynomials and the problem of moments,, SIAM J. Optimization, 11 (): 796. doi: 10.1137/S1052623400366802. 
[21] 
P. Lötstedt, Solving the minimal least squares problem subject to bounds on the variables, BIT, 24 (1984), 206224. 
[22] 
P. Parrilo, "Structured Semidefinite Programs and SemiAlgebraic Geometry Methods in Robustness and Optimization," Ph.D. Thesis, California Institute of Technology, 2000. 
[23] 
J. F. Strum and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246267. doi: 10.1287/moor.28.2.246.14485. 
[24] 
X. Sun, C. Liu, D. Li and J. Gao, On duality gap in binary quadratic programming,, Available from: \url{http://www.optimizationonline.org/DB_FILE/2010/01/2512.pdf}., (). 
[25] 
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. doi: 10.3934/jimo.2008.4.213. 
[26] 
L. F. Zuluage, J. Vera and J. Peña, LMI approximations for cones of positive semidefinite forms, SIAM J. Optimization, 16 (2006), 10761091. 
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