American Institute of Mathematical Sciences

Advanced
January  2009, 5(1): 47-60. doi: 10.3934/jimo.2009.5.47

A Lagrangian dual and surrogate method for multi-dimensional quadratic knapsack problems

Xiaoling Sun 1, , Xiaojin Zheng 2, and  Juan Sun 2,

1. 

School of Management, Fudan University, Shanghai 200433
2. 

Department of Mathematics, Shanghai University, Shanghai 200444, China, China

Received  April 2008 Revised  October 2008 Published  December 2008

Quadratic 0-1 knapsack problems have a variety of applications in various areas such as flexible manufacturing systems, location of transportation facilities and telecommunications. In this paper we present a branch-and-bound method for solving multi-dimensional quadratic 0-1 knapsack problems. Outer approximation and bundle method are used to compute the Lagrangian bound where the Lagrangian relaxation is solved by the maximum flow algorithm. We also present a surrogate constraint heuristic for finding feasible solutions. Preliminary computational results for small to medium size test problems are reported.
Keywords: branch-and-bound method, bundle method., outer approximation, Multi-dimensional quadratic 0-1 knapsack problem.
Mathematics Subject Classification: Primary: 90C10, 90C20; Secondary: 90C1.
Citation: Xiaoling Sun, Xiaojin Zheng, Juan Sun. A Lagrangian dual and surrogate method for multi-dimensional quadratic knapsack problems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 47-60. doi: 10.3934/jimo.2009.5.47
[1]

Jing Zhou, Dejun Chen, Zhenbo Wang, Wenxun Xing. A conic approximation method for the 0-1 quadratic knapsack problem. Journal of Industrial & Management Optimization, 2013, 9 (3) : 531-547. doi: 10.3934/jimo.2013.9.531
[2]

Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A socp relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019104
[3]

Jing Zhou, Zhibin Deng. A low-dimensional SDP relaxation based spatial branch and bound method for nonconvex quadratic programs. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2087-2102. doi: 10.3934/jimo.2019044
[4]

Xiaoling Sun, Hongbo Sheng, Duan Li. An exact algorithm for 0-1 polynomial knapsack problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 223-232. doi: 10.3934/jimo.2007.3.223
[5]

Shu-Cherng Fang, David Y. Gao, Ruey-Lin Sheu, Soon-Yi Wu. Canonical dual approach to solving 0-1 quadratic programming problems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 125-142. doi: 10.3934/jimo.2008.4.125
[6]

Z.G. Feng, K.L. Teo, Y. Zhao. Branch and bound method for sensor scheduling in discrete time. Journal of Industrial & Management Optimization, 2005, 1 (4) : 499-512. doi: 10.3934/jimo.2005.1.499
[7]

Cheng Lu, Zhenbo Wang, Wenxun Xing, Shu-Cherng Fang. Extended canonical duality and conic programming for solving 0-1 quadratic programming problems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 779-793. doi: 10.3934/jimo.2010.6.779
[8]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[9]

Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019135
[10]

S. Kanagawa, K. Inoue, A. Arimoto, Y. Saisho. Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method. Conference Publications, 2005, 2005 (Special) : 463-475. doi: 10.3934/proc.2005.2005.463
[11]

Weipeng Hu, Zichen Deng, Yuyue Qin. Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation. Journal of Geometric Mechanics, 2013, 5 (3) : 295-318. doi: 10.3934/jgm.2013.5.295
[12]

Wen-ling Zhao, Dao-jin Song. A global error bound via the SQP method for constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 775-781. doi: 10.3934/jimo.2007.3.775
[13]

Tatsien Li, Wancheng Sheng. The general multi-dimensional Riemann problem for hyperbolic systems with real constant coefficients. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 737-744. doi: 10.3934/dcds.2002.8.737
[14]

Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057
[15]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
[16]

Kang-Ling Liao, Chih-Wen Shih, Chi-Jer Yu. The snapback repellers for chaos in multi-dimensional maps. Journal of Computational Dynamics, 2018, 5 (1&2) : 81-92. doi: 10.3934/jcd.2018004
[17]

Franz Achleitner, Anton Arnold, Eric A. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic & Related Models, 2018, 11 (4) : 953-1009. doi: 10.3934/krm.2018038
[18]

Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652
[19]

Gerald Sommer, Di Zang. Parity symmetry in multi-dimensional signals. Communications on Pure & Applied Analysis, 2007, 6 (3) : 829-852. doi: 10.3934/cpaa.2007.6.829
[20]

Yuzhong Zhang, Fan Zhang, Maocheng Cai. Some new results on multi-dimension Knapsack problem. Journal of Industrial & Management Optimization, 2005, 1 (3) : 315-321. doi: 10.3934/jimo.2005.1.315

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences