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Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems
Manifold relaxations for integer programming
1. | College of Mathematics, Chongqing Normal University, Chongqing, China |
2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
References:
[1] |
M. Borchardt, An exact penalty approach for solving a class of minimization problems with Boolean variables,, Optimization, 19 (1988), 829.
doi: 10.1080/02331938808843396. |
[2] |
Q. Chai, R. Loxton, K. L. Teo and C. H. Yang, A max-min control problem arising in gradient elution chromatography,, Ind. Eng. Chem. Res., 51 (2012), 6137.
doi: 10.1021/ie202475p. |
[3] |
A. Edelman, T. A. Arias and S. Smith, The geometry of algorithms with orthogonality constraints,, SIAM J. Matrix Anal. Appl., 20 (1998), 303.
doi: 10.1137/S0895479895290954. |
[4] |
Z. G. Feng and K. L. Teo, A discrete filled function method for the design of FIR filters with signed-powers-of-two coefficients,, IEEE Trans. on Signal Process., 56 (2008), 134.
doi: 10.1109/TSP.2007.901164. |
[5] |
Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time,, J. Ind. Manag. Optim., 1 (2005), 499.
doi: 10.3934/jimo.2005.1.499. |
[6] |
R. Fletcher, Practical Methods of Optimization,, 2nd edition, (1987).
|
[7] |
C. Helmberg and F. Rendl, Solving quadratic (0,1)-problems by semidefinite programming and cutting planes,, Math. Programming, 82 (1998), 291.
doi: 10.1007/BF01580072. |
[8] |
M. Jünger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, Giovanni Rinaldi and L. A. Wolsey, eds., 50 Years of Integer Programming 1958-2008. From the Early Years to the State-of-the-Art,, Papers from the 12th Combinatorial Optimization Workshop (AUSSOIS 2008) held in Aussois, (2008), 7.
doi: 10.1007/978-3-540-68279-0. |
[9] |
B. Kalantari and J. B. Rosen, Penalty formulation for zero-one nonlinear programming,, Discrete Appl. Math., 16 (1987), 179.
doi: 10.1016/0166-218X(87)90073-4. |
[10] |
W. Murray and K.-M. Ng, An algorithm for nonlinear optimization problems with binary variables,, Comput. Optim. Appl., 47 (2010), 257.
doi: 10.1007/s10589-008-9218-1. |
[11] |
C.-K. Ng, L.-S. Zhang, D. Li and W.-W. Tian, Discrete filled function method for discrete global optimization,, Comput. Optim. Appl., 31 (2005), 87.
doi: 10.1007/s10589-005-0985-7. |
[12] |
P. M. Pardalos, O. A. Prokopyev and S. Busygin, Continuous approaches for solving discrete optimization problems,, in Handbook on Modelling for Discrete Optimization, (2006), 39.
doi: 10.1007/0-387-32942-0_2. |
[13] |
J. Richstein, Verifying the Goldbach conjecture up to $4\cdot 10^{14}$,, Math. Comp., 70 (2001), 1745.
doi: 10.1090/S0025-5718-00-01290-4. |
[14] |
K. Schittkowski, More Test Examples for Nonlinear Programming Codes,, Lecture Notes in Economics and Mathematical Systems, (1987).
doi: 10.1007/978-3-642-61582-5. |
[15] |
R. A. Shandiz and N. Mahdavi-amiri, An exact penalty approach for mixed integer nonlinear programming problems,, American Journal of Operations Research, 1 (2011), 185. Google Scholar |
[16] |
H. D. Sherali and W. P. Adams, A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems,, Discrete Appl. Math., 52 (1994), 83.
doi: 10.1016/0166-218X(92)00190-W. |
[17] |
S. Wang, K. L. Teo, H. W. J. Lee and L. Caccetta, Solving 0-1 programming problems by a penalty approach,, Opsearch, 34 (1997), 196.
|
[18] |
W.-Y. Yan and K. L. Teo, Optimal finite-precision approximation of FIR filters,, Signal Processing, 82 (2002), 1695.
doi: 10.1016/S0165-1684(02)00331-6. |
[19] |
K. F. C Yiu, Y. Liu and K. L. Teo, A hybrid descent method for global optimization,, J. Global Optim., 28 (2004), 229.
doi: 10.1023/B:JOGO.0000015313.93974.b0. |
[20] |
K. F. C Yiu, W. Y. Yan, K. L. Teo and S. Y. Low, A new hybrid descent method with application to the optimal design of finite precision FIR filters,, Optim. Methods Softw., 25 (2010), 725.
doi: 10.1080/10556780903254104. |
[21] |
C. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation,, Journal of Global Optimization, 56 (2013), 503.
doi: 10.1007/s10898-012-9858-7. |
[22] |
W. X. Zhu, Penalty parameter for linearly constrained 0-1 quadratic programming,, J. Optim. Theory Appl., 116 (2003), 229.
doi: 10.1023/A:1022174505886. |
show all references
References:
[1] |
M. Borchardt, An exact penalty approach for solving a class of minimization problems with Boolean variables,, Optimization, 19 (1988), 829.
doi: 10.1080/02331938808843396. |
[2] |
Q. Chai, R. Loxton, K. L. Teo and C. H. Yang, A max-min control problem arising in gradient elution chromatography,, Ind. Eng. Chem. Res., 51 (2012), 6137.
doi: 10.1021/ie202475p. |
[3] |
A. Edelman, T. A. Arias and S. Smith, The geometry of algorithms with orthogonality constraints,, SIAM J. Matrix Anal. Appl., 20 (1998), 303.
doi: 10.1137/S0895479895290954. |
[4] |
Z. G. Feng and K. L. Teo, A discrete filled function method for the design of FIR filters with signed-powers-of-two coefficients,, IEEE Trans. on Signal Process., 56 (2008), 134.
doi: 10.1109/TSP.2007.901164. |
[5] |
Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time,, J. Ind. Manag. Optim., 1 (2005), 499.
doi: 10.3934/jimo.2005.1.499. |
[6] |
R. Fletcher, Practical Methods of Optimization,, 2nd edition, (1987).
|
[7] |
C. Helmberg and F. Rendl, Solving quadratic (0,1)-problems by semidefinite programming and cutting planes,, Math. Programming, 82 (1998), 291.
doi: 10.1007/BF01580072. |
[8] |
M. Jünger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, Giovanni Rinaldi and L. A. Wolsey, eds., 50 Years of Integer Programming 1958-2008. From the Early Years to the State-of-the-Art,, Papers from the 12th Combinatorial Optimization Workshop (AUSSOIS 2008) held in Aussois, (2008), 7.
doi: 10.1007/978-3-540-68279-0. |
[9] |
B. Kalantari and J. B. Rosen, Penalty formulation for zero-one nonlinear programming,, Discrete Appl. Math., 16 (1987), 179.
doi: 10.1016/0166-218X(87)90073-4. |
[10] |
W. Murray and K.-M. Ng, An algorithm for nonlinear optimization problems with binary variables,, Comput. Optim. Appl., 47 (2010), 257.
doi: 10.1007/s10589-008-9218-1. |
[11] |
C.-K. Ng, L.-S. Zhang, D. Li and W.-W. Tian, Discrete filled function method for discrete global optimization,, Comput. Optim. Appl., 31 (2005), 87.
doi: 10.1007/s10589-005-0985-7. |
[12] |
P. M. Pardalos, O. A. Prokopyev and S. Busygin, Continuous approaches for solving discrete optimization problems,, in Handbook on Modelling for Discrete Optimization, (2006), 39.
doi: 10.1007/0-387-32942-0_2. |
[13] |
J. Richstein, Verifying the Goldbach conjecture up to $4\cdot 10^{14}$,, Math. Comp., 70 (2001), 1745.
doi: 10.1090/S0025-5718-00-01290-4. |
[14] |
K. Schittkowski, More Test Examples for Nonlinear Programming Codes,, Lecture Notes in Economics and Mathematical Systems, (1987).
doi: 10.1007/978-3-642-61582-5. |
[15] |
R. A. Shandiz and N. Mahdavi-amiri, An exact penalty approach for mixed integer nonlinear programming problems,, American Journal of Operations Research, 1 (2011), 185. Google Scholar |
[16] |
H. D. Sherali and W. P. Adams, A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems,, Discrete Appl. Math., 52 (1994), 83.
doi: 10.1016/0166-218X(92)00190-W. |
[17] |
S. Wang, K. L. Teo, H. W. J. Lee and L. Caccetta, Solving 0-1 programming problems by a penalty approach,, Opsearch, 34 (1997), 196.
|
[18] |
W.-Y. Yan and K. L. Teo, Optimal finite-precision approximation of FIR filters,, Signal Processing, 82 (2002), 1695.
doi: 10.1016/S0165-1684(02)00331-6. |
[19] |
K. F. C Yiu, Y. Liu and K. L. Teo, A hybrid descent method for global optimization,, J. Global Optim., 28 (2004), 229.
doi: 10.1023/B:JOGO.0000015313.93974.b0. |
[20] |
K. F. C Yiu, W. Y. Yan, K. L. Teo and S. Y. Low, A new hybrid descent method with application to the optimal design of finite precision FIR filters,, Optim. Methods Softw., 25 (2010), 725.
doi: 10.1080/10556780903254104. |
[21] |
C. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation,, Journal of Global Optimization, 56 (2013), 503.
doi: 10.1007/s10898-012-9858-7. |
[22] |
W. X. Zhu, Penalty parameter for linearly constrained 0-1 quadratic programming,, J. Optim. Theory Appl., 116 (2003), 229.
doi: 10.1023/A:1022174505886. |
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