American Institute of Mathematical Sciences

Advanced
April  2007, 3(2): 293-304. doi: 10.3934/jimo.2007.3.293

Solutions and optimality criteria to box constrained nonconvex minimization problems

David Yang Gao 1,

1. 

Department of Mathematics & Grado, Department of Industrial and System Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, United States

Received  August 2006 Revised  January 2007 Published  April 2007

This paper presents a canonical duality theory for solving nonconvex polynomial programming problems subjected to box constraints. It is proved that under certain conditions, the constrained nonconvex problems can be converted to the so-called canonical (perfect) dual problems, which can be solved by deterministic methods. Both global and local extrema of the primal problems can be identified by a triality theory proposed by the author. Applications to nonconvex integer programming and Boolean least squares problems are discussed. Examples are illustrated. A conjecture on NP-hard problems is proposed.
Keywords: NP-hard problems., Global optimization; Duality; Nonconvex minimization; Box constraints; Integer programming; Boolean least squares problem.
Mathematics Subject Classification: 49N15, 49M37, 90C26, 90C2.
Citation: David Yang Gao. Solutions and optimality criteria to box constrained nonconvex minimization problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 293-304. doi: 10.3934/jimo.2007.3.293
[1]

David Yang Gao. Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints. Journal of Industrial & Management Optimization, 2005, 1 (1) : 53-63. doi: 10.3934/jimo.2005.1.53
[2]

Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027
[3]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019102
[4]

Mila Nikolova. Analytical bounds on the minimizers of (nonconvex) regularized least-squares. Inverse Problems & Imaging, 2008, 2 (1) : 133-149. doi: 10.3934/ipi.2008.2.133
[5]

Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67
[6]

Hassan Mohammad, Mohammed Yusuf Waziri, Sandra Augusta Santos. A brief survey of methods for solving nonlinear least-squares problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 1-13. doi: 10.3934/naco.2019001
[7]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407
[8]

Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial & Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177
[9]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019033
[10]

Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415
[11]

Zhenbo Wang, Shu-Cherng Fang, David Y. Gao, Wenxun Xing. Global extremal conditions for multi-integer quadratic programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 213-225. doi: 10.3934/jimo.2008.4.213
[12]

Mohamed A. Tawhid, Ahmed F. Ali. A simplex grey wolf optimizer for solving integer programming and minimax problems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 301-323. doi: 10.3934/naco.2017020
[13]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial & Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523
[14]

Zhou Sheng, Gonglin Yuan, Zengru Cui, Xiabin Duan, Xiaoliang Wang. An adaptive trust region algorithm for large-residual nonsmooth least squares problems. Journal of Industrial & Management Optimization, 2018, 14 (2) : 707-718. doi: 10.3934/jimo.2017070
[15]

H. D. Scolnik, N. E. Echebest, M. T. Guardarucci. Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 175-191. doi: 10.3934/jimo.2009.5.175
[16]

K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026
[17]

Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 2 (2) : 107-126. doi: 10.3934/mfc.2019009
[18]

Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070
[19]

Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial & Management Optimization, 2010, 6 (3) : 497-500. doi: 10.3934/jimo.2010.6.497
[20]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (41)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences