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An alternating linearization method with inexact data for bilevel nonsmooth convex optimization
Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems
1. | Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China |
2. | Department of Management Science and Engineering, Zhejiang University, Hangzhou, Zhejiang 310058, China |
References:
[1] |
I. M. Bomze, Copositivity conditions for global optimality in indefinite quadratic programming problems, Czechosl. J. Oper. Res., 1 (1992), 7-19. |
[2] |
I. M. Bomze, Copositive optimization - recent developments and applications, Eur. J. Oper. Res., 216 (2012), 509-520.
doi: 10.1016/j.ejor.2011.04.026. |
[3] |
Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, Eur. J. Oper. Res., 229 (2013), 21-28.
doi: 10.1016/j.ejor.2013.02.031. |
[4] |
M. Dür, Copositive programming - a survey, in Recent Advances in Optimization and its Applications in Engineering, Springer, 2010, 3-20. |
[5] |
D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optim., 17 (2000), 127-160.
doi: 10.1023/A:1026537630859. |
[6] |
D. Y. Gao, Advances in canonical duality theory with applications to global optimization, in Proceedings of the Fifth International Conference on Foundations of Computer-Aided Process Operations (FOCAPO 2008) (eds. M. Ierapetriou, M. Bassett and S. Pistikopoulos), Omni Press, 2008, 73-82. |
[7] |
M. R. Garey and D. S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, CA, 1979. |
[8] |
J.-B. Hiriart-Urruty and A. Seeger, Variational approach to copositive matrices, SIAM Rev., 52 (2010), 593-629.
doi: 10.1137/090750391. |
[9] |
V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions, Math. Program., 110 (2007), 521-541.
doi: 10.1007/s10107-006-0012-5. |
[10] |
J. Linderoth, A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Math. Program. Ser. B, 103 (2005), 251-282.
doi: 10.1007/s10107-005-0582-7. |
[11] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Algebra Appl., 284 (1998), 193-228.
doi: 10.1016/S0024-3795(98)10032-0. |
[12] |
C. Lu, S.-C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, SIAM J. Optim., 21 (2011), 1475-1490.
doi: 10.1137/100793955. |
[13] |
C. Lu, Z. Wang, W. Xing and S.-C. Fang, Extended canonical duality and conic programming for solving 0-1 quadratic programming problems, J. Ind. Manag. Optim., 6 (2010), 779-793.
doi: 10.3934/jimo.2010.6.779. |
[14] |
P. M. Pardalos and S. A. Vavasis, Quadratic programming with one negative eigenvalue is NP-hard, J. Global Optim., 1 (1991), 15-22.
doi: 10.1007/BF00120662. |
[15] |
J.-M. Peng and Y.-X. Yuan, Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM J. Optim., 7 (1997), 579-594.
doi: 10.1137/S1052623494261520. |
[16] |
A. Saxena, P. Bonami and J. Lee, Convex relaxations of non-convex mixed integer quadratically constrained programs: Extended formulations, Math. Program. Ser. B, 124 (2010), 383-411.
doi: 10.1007/s10107-010-0371-9. |
[17] |
J. F. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Math. Oper. Res., 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
[18] |
J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods, Optim. Methods Softw., 11/12 (1999), 625-653.
doi: 10.1080/10556789908805766. |
[19] |
Y. Tian, S.-C. Fang, Z. Deng and W. Xing, Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming, J. Ind. Manag. Optim., 9 (2013), 703-721.
doi: 10.3934/jimo.2013.9.703. |
[20] |
Y. Tian and C. Lu, Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems, J. Ind. Manag. Optim., 7 (2011), 1027-1039.
doi: 10.3934/jimo.2011.7.1027. |
[21] |
J. Zhou, D. Chen, Z. Wang and W. Xing, A conic approximation method for the 0-1 quadratic knapsack problem, J. Ind. Manag. Optim., 9 (2013), 531-547.
doi: 10.3934/jimo.2013.9.531. |
show all references
References:
[1] |
I. M. Bomze, Copositivity conditions for global optimality in indefinite quadratic programming problems, Czechosl. J. Oper. Res., 1 (1992), 7-19. |
[2] |
I. M. Bomze, Copositive optimization - recent developments and applications, Eur. J. Oper. Res., 216 (2012), 509-520.
doi: 10.1016/j.ejor.2011.04.026. |
[3] |
Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, Eur. J. Oper. Res., 229 (2013), 21-28.
doi: 10.1016/j.ejor.2013.02.031. |
[4] |
M. Dür, Copositive programming - a survey, in Recent Advances in Optimization and its Applications in Engineering, Springer, 2010, 3-20. |
[5] |
D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optim., 17 (2000), 127-160.
doi: 10.1023/A:1026537630859. |
[6] |
D. Y. Gao, Advances in canonical duality theory with applications to global optimization, in Proceedings of the Fifth International Conference on Foundations of Computer-Aided Process Operations (FOCAPO 2008) (eds. M. Ierapetriou, M. Bassett and S. Pistikopoulos), Omni Press, 2008, 73-82. |
[7] |
M. R. Garey and D. S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, CA, 1979. |
[8] |
J.-B. Hiriart-Urruty and A. Seeger, Variational approach to copositive matrices, SIAM Rev., 52 (2010), 593-629.
doi: 10.1137/090750391. |
[9] |
V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions, Math. Program., 110 (2007), 521-541.
doi: 10.1007/s10107-006-0012-5. |
[10] |
J. Linderoth, A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Math. Program. Ser. B, 103 (2005), 251-282.
doi: 10.1007/s10107-005-0582-7. |
[11] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Algebra Appl., 284 (1998), 193-228.
doi: 10.1016/S0024-3795(98)10032-0. |
[12] |
C. Lu, S.-C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, SIAM J. Optim., 21 (2011), 1475-1490.
doi: 10.1137/100793955. |
[13] |
C. Lu, Z. Wang, W. Xing and S.-C. Fang, Extended canonical duality and conic programming for solving 0-1 quadratic programming problems, J. Ind. Manag. Optim., 6 (2010), 779-793.
doi: 10.3934/jimo.2010.6.779. |
[14] |
P. M. Pardalos and S. A. Vavasis, Quadratic programming with one negative eigenvalue is NP-hard, J. Global Optim., 1 (1991), 15-22.
doi: 10.1007/BF00120662. |
[15] |
J.-M. Peng and Y.-X. Yuan, Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM J. Optim., 7 (1997), 579-594.
doi: 10.1137/S1052623494261520. |
[16] |
A. Saxena, P. Bonami and J. Lee, Convex relaxations of non-convex mixed integer quadratically constrained programs: Extended formulations, Math. Program. Ser. B, 124 (2010), 383-411.
doi: 10.1007/s10107-010-0371-9. |
[17] |
J. F. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Math. Oper. Res., 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
[18] |
J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods, Optim. Methods Softw., 11/12 (1999), 625-653.
doi: 10.1080/10556789908805766. |
[19] |
Y. Tian, S.-C. Fang, Z. Deng and W. Xing, Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming, J. Ind. Manag. Optim., 9 (2013), 703-721.
doi: 10.3934/jimo.2013.9.703. |
[20] |
Y. Tian and C. Lu, Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems, J. Ind. Manag. Optim., 7 (2011), 1027-1039.
doi: 10.3934/jimo.2011.7.1027. |
[21] |
J. Zhou, D. Chen, Z. Wang and W. Xing, A conic approximation method for the 0-1 quadratic knapsack problem, J. Ind. Manag. Optim., 9 (2013), 531-547.
doi: 10.3934/jimo.2013.9.531. |
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