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Quadratic optimization over a polyhedral cone
1. | School of Business Administration, Southwestern University of Finance and Economics, Chengdu, 611130 |
2. | Department of Management Science and Engineering, Zhejiang University, Hangzhou, Zhejiang 310058 |
3. | School of Management, University of Chinese Academy of Sciences, Beijing, 100190, China |
References:
[1] |
L. Angulo-Meza and M. Lins, Review of methods for increasing discrimination in data envelopment analysis, Annals of Operations Research, 116 (2002), 225-242.
doi: 10.1023/A:1021340616758. |
[2] |
K. Anstreicher, Semidefinite programming versus the Reformulation-Linearization Technique for nonconvex quadratically constrained quadratic programming, Journal of Global Optimization, 43 (2009), 471-484.
doi: 10.1007/s10898-008-9372-0. |
[3] |
A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization Analysis, Algorithms and Engineering Applications, SIAM, Philadelphis, 2001.
doi: 10.1137/1.9780898718829. |
[4] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441. |
[5] |
S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495.
doi: 10.1007/s10107-008-0223-z. |
[6] |
Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, European Journal of Operational Research, 229 (2013), 21-28.
doi: 10.1016/j.ejor.2013.02.031. |
[7] |
M. Grant and S. Boyd, CVX: matlab Software for Disciplined Programming, version 1.2, 2010. http://cvxr.com/cvx |
[8] |
X. Guo, Z. Deng, S.-C. Fang and W. Xing, Quadratic optimization over one first-order cone, Journal of Industrial and Management Optimization, 10 (2014), 945-963.
doi: 10.3934/jimo.2014.10.945. |
[9] |
P. Hansen, B. Jaumard, M. Ruiz and J. Xiong, Global minimization of indefinite quadratic functions subject to box constraints, Naval Research Logistics, 40 (1993), 373-392.
doi: 10.1002/1520-6750(199304)40:3<373::AID-NAV3220400307>3.0.CO;2-A. |
[10] |
J. Hiriat-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Berlag, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[11] |
H. Jiang, M. Fukushima, L. Qi and D. Sun, A trust region method for solving generalized complementarity problem, SIAM Journal on Optimization, 8 (1998), 140-157.
doi: 10.1137/S1052623495296541. |
[12] |
Q. Jin, Y. Tian, Z. Deng, S.-C. Fang and W. Xing, Exact computable representation of some second-order cone constrained quadratic programming problems, Journal of Operations Research Society of China, 1 (2013), 107-134.
doi: 10.1007/s40305-013-0009-8. |
[13] |
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 Journal on Optimization, 21 (2010), 1475-1490.
doi: 10.1137/100793955. |
[14] |
H. Kunze and D. Siegel, A graph theoretic approach to strong monotonicity with respect to polyhedral cones, Positivity, 6 (2002), 95-113.
doi: 10.1023/A:1015290601993. |
[15] |
F. Ma, G. Sheng and Y. Yin, A superlinearly convergent method for the generalized complementarity problem over a polyhedral cone, Journal of Applied Mathematics, (2013), Art. ID 671402, 6 pp.
doi: 10.1155/2013/671402. |
[16] |
R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1997. |
[17] |
A. Sinha, P. Korhonen, J. Wallenius and K. Deb, An interactive evolutionary multi-objective optimization method based on polyhedral cones, Learning and Intelligent Optimization, 6073 (2010), 318-332.
doi: 10.1007/978-3-642-13800-3_33. |
[18] |
J. Stoer and C. Witzgall, Convexity and Optimization In Finite Dimensions, Springer-Berlag Berlin, 1970. |
[19] |
J. Sturm, SeDuMi 1.02, a matlab tool box for optimization over symmetric cones, Optimization Methods and Software, 11 & 12 (1999), 625-653.
doi: 10.1080/10556789908805766. |
[20] |
J. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
[21] |
H. Sun and Y. Wang, Further discussion on the error bound for generalized linear complementarity problem over a polyhedral cone, Journal of Optimization and Theory Application, 159 (2013), 93-107.
doi: 10.1007/s10957-013-0290-z. |
[22] |
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 positiv programming, Journal of Industrial and Management Optimization, 9 (2013), 703-721.
doi: 10.3934/jimo.2013.9.703. |
[23] |
Y. Wang, F. Ma and J. Zhang, A nonsmooth L-M method for solving the generalized nonlinear complementarity problem over a polyhedral cone, Applied Mathematics and Optimization, 52 (2005), 73-92.
doi: 10.1007/s00245-005-0823-4. |
[24] |
Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267.
doi: 10.1137/S105262340139001X. |
show all references
References:
[1] |
L. Angulo-Meza and M. Lins, Review of methods for increasing discrimination in data envelopment analysis, Annals of Operations Research, 116 (2002), 225-242.
doi: 10.1023/A:1021340616758. |
[2] |
K. Anstreicher, Semidefinite programming versus the Reformulation-Linearization Technique for nonconvex quadratically constrained quadratic programming, Journal of Global Optimization, 43 (2009), 471-484.
doi: 10.1007/s10898-008-9372-0. |
[3] |
A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization Analysis, Algorithms and Engineering Applications, SIAM, Philadelphis, 2001.
doi: 10.1137/1.9780898718829. |
[4] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441. |
[5] |
S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495.
doi: 10.1007/s10107-008-0223-z. |
[6] |
Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, European Journal of Operational Research, 229 (2013), 21-28.
doi: 10.1016/j.ejor.2013.02.031. |
[7] |
M. Grant and S. Boyd, CVX: matlab Software for Disciplined Programming, version 1.2, 2010. http://cvxr.com/cvx |
[8] |
X. Guo, Z. Deng, S.-C. Fang and W. Xing, Quadratic optimization over one first-order cone, Journal of Industrial and Management Optimization, 10 (2014), 945-963.
doi: 10.3934/jimo.2014.10.945. |
[9] |
P. Hansen, B. Jaumard, M. Ruiz and J. Xiong, Global minimization of indefinite quadratic functions subject to box constraints, Naval Research Logistics, 40 (1993), 373-392.
doi: 10.1002/1520-6750(199304)40:3<373::AID-NAV3220400307>3.0.CO;2-A. |
[10] |
J. Hiriat-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Berlag, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[11] |
H. Jiang, M. Fukushima, L. Qi and D. Sun, A trust region method for solving generalized complementarity problem, SIAM Journal on Optimization, 8 (1998), 140-157.
doi: 10.1137/S1052623495296541. |
[12] |
Q. Jin, Y. Tian, Z. Deng, S.-C. Fang and W. Xing, Exact computable representation of some second-order cone constrained quadratic programming problems, Journal of Operations Research Society of China, 1 (2013), 107-134.
doi: 10.1007/s40305-013-0009-8. |
[13] |
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 Journal on Optimization, 21 (2010), 1475-1490.
doi: 10.1137/100793955. |
[14] |
H. Kunze and D. Siegel, A graph theoretic approach to strong monotonicity with respect to polyhedral cones, Positivity, 6 (2002), 95-113.
doi: 10.1023/A:1015290601993. |
[15] |
F. Ma, G. Sheng and Y. Yin, A superlinearly convergent method for the generalized complementarity problem over a polyhedral cone, Journal of Applied Mathematics, (2013), Art. ID 671402, 6 pp.
doi: 10.1155/2013/671402. |
[16] |
R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1997. |
[17] |
A. Sinha, P. Korhonen, J. Wallenius and K. Deb, An interactive evolutionary multi-objective optimization method based on polyhedral cones, Learning and Intelligent Optimization, 6073 (2010), 318-332.
doi: 10.1007/978-3-642-13800-3_33. |
[18] |
J. Stoer and C. Witzgall, Convexity and Optimization In Finite Dimensions, Springer-Berlag Berlin, 1970. |
[19] |
J. Sturm, SeDuMi 1.02, a matlab tool box for optimization over symmetric cones, Optimization Methods and Software, 11 & 12 (1999), 625-653.
doi: 10.1080/10556789908805766. |
[20] |
J. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
[21] |
H. Sun and Y. Wang, Further discussion on the error bound for generalized linear complementarity problem over a polyhedral cone, Journal of Optimization and Theory Application, 159 (2013), 93-107.
doi: 10.1007/s10957-013-0290-z. |
[22] |
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 positiv programming, Journal of Industrial and Management Optimization, 9 (2013), 703-721.
doi: 10.3934/jimo.2013.9.703. |
[23] |
Y. Wang, F. Ma and J. Zhang, A nonsmooth L-M method for solving the generalized nonlinear complementarity problem over a polyhedral cone, Applied Mathematics and Optimization, 52 (2005), 73-92.
doi: 10.1007/s00245-005-0823-4. |
[24] |
Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267.
doi: 10.1137/S105262340139001X. |
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