American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method
  • JIMO Home
  • This Issue
  • Next Article
    The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains
January  2010, 6(1): 177-196. doi: 10.3934/jimo.2010.6.177

Global optimization algorithm for solving bilevel programming problems with quadratic lower levels

Paul B. Hermanns 1, and  Nguyen Van Thoai 1,

1. 

Department of Mathematics, University of Trier, 54286 Trier, Germany, Germany

Received  March 2009 Revised  September 2009 Published  November 2009

In this article, we propose a method for finding the global optimum of a class of nonlinear bilevel programming problems. The main idea of this method is to construct iteratively a sequence of points either ending at an optimal solution of the equivalent problem with a complementarity constraint, or converging to an optimal solution. The construction of such a sequence is performed by using a branch-and-bound scheme, together with some relaxation techniques, which are successfully applied in global optimization. Some illustrative examples and results on computational experiments are reported.
Keywords: Bilevel programming, nonconvex programming, branch and bound..
Mathematics Subject Classification: Primary: 90C26; Secondary: 91A0.
Citation: Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial and Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177
[1]

Jing Zhou, Zhibin Deng. A low-dimensional SDP relaxation based spatial branch and bound method for nonconvex quadratic programs. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2087-2102. doi: 10.3934/jimo.2019044
[2]

Yue Zheng, Zhongping Wan, Shihui Jia, Guangmin Wang. A new method for strong-weak linear bilevel programming problem. Journal of Industrial and Management Optimization, 2015, 11 (2) : 529-547. doi: 10.3934/jimo.2015.11.529
[3]

Bhawna Kohli. Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3209-3221. doi: 10.3934/jimo.2020114
[4]

Le Thi Hoai An, Tran Duc Quynh, Pham Dinh Tao. A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 167-185. doi: 10.3934/naco.2012.2.167
[5]

Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027
[6]

Dan Xue, Wenyu Sun, Hongjin He. A structured trust region method for nonconvex programming with separable structure. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 283-293. doi: 10.3934/naco.2013.3.283
[7]

Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111
[8]

Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102
[9]

Nazih Abderrazzak Gadhi. A note on the paper "Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem". Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021103
[10]

Ziye Shi, Qingwei Jin. Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 871-882. doi: 10.3934/jimo.2014.10.871
[11]

Yang Li, Yonghong Ren, Yun Wang, Jian Gu. Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved. Journal of Industrial and Management Optimization, 2015, 11 (1) : 65-81. doi: 10.3934/jimo.2015.11.65
[12]

Yang Li, Liwei Zhang. A nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming. Journal of Industrial and Management Optimization, 2009, 5 (3) : 651-669. doi: 10.3934/jimo.2009.5.651
[13]

Z.G. Feng, K.L. Teo, Y. Zhao. Branch and bound method for sensor scheduling in discrete time. Journal of Industrial and Management Optimization, 2005, 1 (4) : 499-512. doi: 10.3934/jimo.2005.1.499
[14]

Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial and Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647
[15]

Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial and Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557
[16]

Qinghong Zhang, Gang Chen, Ting Zhang. Duality formulations in semidefinite programming. Journal of Industrial and Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881
[17]

Charles Fefferman. Interpolation by linear programming I. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477
[18]

Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193
[19]

Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial and Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104
[20]

Andrzej Nowakowski, Jan Sokolowski. On dual dynamic programming in shape control. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2473-2485. doi: 10.3934/cpaa.2012.11.2473

2021 Impact Factor: 1.411

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy