Manifold relaxations for integer programming

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April 2014, 10(2): 557-566. doi: 10.3934/jimo.2014.10.557

Manifold relaxations for integer programming

Zhiguo Feng ^1, and Ka-Fai Cedric Yiu ^2,

1.	College of Mathematics, Chongqing Normal University, Chongqing, China
2.	Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received October 2012 Revised August 2013 Published October 2013

Abstract
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In this paper, the integer programming problem is studied. We introduce the notion of integer basis and show that a given integer set can be converted into a fixed number of linear combinations of the basis elements. By employing the Stiefel manifold and optimal control theory, the combinatorial optimization problem can be converted into a continuous optimization problem over the continuous Stiefel manifold. As a result, gradient descent methods can be applied to find the optimal integer solution. We demonstrate by numerical examples that this approach can obtain good solutions. Furthermore, this method gives new insights into continuous relaxation for solving integer programming problems.

Keywords: integer basis, optimal control, Integer programming, Stiefel manifold, permutation matrix..

Mathematics Subject Classification: Primary: 90C10, 90C57; Secondary: 58D1.

Citation: Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	R. Fletcher, Practical Methods of Optimization,, 2nd edition, (1987). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	R. Fletcher, Practical Methods of Optimization,, 2nd edition, (1987). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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