American Institute of Mathematical Sciences

Advanced
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

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

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.  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

[1]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[2]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[3]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[4]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[5]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[6]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[7]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[8]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179
[9]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[10]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[11]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[12]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051
[13]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032
[14]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[15]

Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053
[16]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031
[17]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032
[18]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
[19]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[20]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences