2011, 7(1): 103-115. doi: 10.3934/jimo.2011.7.103

Multistage hierarchical optimization problems with multi-criterion objectives

1. 

Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI 49855, United States, United States

Received  October 2009 Revised  October 2010 Published  January 2011

A hierarchical optimization (or bilevel programming) problem consists of a decision maker called the leader who is interested in optimizing an objective function that involves with the decisions from another decision maker called the follower whose decisions are based in part on the policies made by the leader. However, if the planning horizon expands into an extended period of time, it may be unrealistic for either players to commit to the original decisions so there is a desire to break the problem into stages and the leader may wish to reevaluate the follower's response at each stage. In this article, we propose a multistage hierarchical optimization problem with the leader's objective consisting of multiple criteria and study the optimality conditions of such problems using an extremal principle of Mordukhovich.
Citation: Roxin Zhang, Bao Truong, Qinghong Zhang. Multistage hierarchical optimization problems with multi-criterion objectives. Journal of Industrial & Management Optimization, 2011, 7 (1) : 103-115. doi: 10.3934/jimo.2011.7.103
References:
[1]

T. Bao, P. Gupta and B. Mordukhovich, Necessary conditions in multiobjective optimization with equilibrium constraints,, Journal of Optimization Theory and Applications, 135 (2007), 179. doi: 10.1007/s10957-007-9209-x.

[2]

S. Dempe, J. Dutta and B. Mordukhovich, New necessary optimality conditions in optimistic bilevel programming,, Optimization, 56 (2007), 577. doi: 10.1080/02331930701617551.

[3]

R. Henrion, B. Mordukhovich and N. Nam, Second-order analysis of polyhedral systems in finite and infinite dimension with applications to robust stability of variational inequalities,, SIAM J. Optim., 20 (2009), 2199. doi: 10.1137/090766413.

[4]

M. Kočvara, M. Kružìk and J. Outrata, On the control of an evolutionary equilibrium in micromagnetics,, in, (2006), 143. doi: 10.1007/0-387-34221-4_8.

[5]

M. Kočvara and J. Outrata, On the modeling and control of delamination processes,, in, (2004), 171.

[6]

B. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory,", Springer, (2006).

[7]

B. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. 2: Applications,", Springer, (2006).

[8]

B. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints,, J. Appl. Math. Mech., 40 (1976), 960. doi: 10.1016/0021-8928(76)90136-2.

[9]

R. Rockafellar and R. Wets, "Variational Analysis,", Springer, (1998). doi: 10.1007/978-3-642-02431-3.

[10]

G. Tzeng, S. Tsau and J. Chen, Application of hierarchy multistage-multiobjective approach to network design: case of express road,, in, (1995), 370.

[11]

R. Zhang, Multistage bilevel programming problems,, Optimization, 52 (2003), 605. doi: 10.1080/02331930310001611420.

show all references

References:
[1]

T. Bao, P. Gupta and B. Mordukhovich, Necessary conditions in multiobjective optimization with equilibrium constraints,, Journal of Optimization Theory and Applications, 135 (2007), 179. doi: 10.1007/s10957-007-9209-x.

[2]

S. Dempe, J. Dutta and B. Mordukhovich, New necessary optimality conditions in optimistic bilevel programming,, Optimization, 56 (2007), 577. doi: 10.1080/02331930701617551.

[3]

R. Henrion, B. Mordukhovich and N. Nam, Second-order analysis of polyhedral systems in finite and infinite dimension with applications to robust stability of variational inequalities,, SIAM J. Optim., 20 (2009), 2199. doi: 10.1137/090766413.

[4]

M. Kočvara, M. Kružìk and J. Outrata, On the control of an evolutionary equilibrium in micromagnetics,, in, (2006), 143. doi: 10.1007/0-387-34221-4_8.

[5]

M. Kočvara and J. Outrata, On the modeling and control of delamination processes,, in, (2004), 171.

[6]

B. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory,", Springer, (2006).

[7]

B. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. 2: Applications,", Springer, (2006).

[8]

B. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints,, J. Appl. Math. Mech., 40 (1976), 960. doi: 10.1016/0021-8928(76)90136-2.

[9]

R. Rockafellar and R. Wets, "Variational Analysis,", Springer, (1998). doi: 10.1007/978-3-642-02431-3.

[10]

G. Tzeng, S. Tsau and J. Chen, Application of hierarchy multistage-multiobjective approach to network design: case of express road,, in, (1995), 370.

[11]

R. Zhang, Multistage bilevel programming problems,, Optimization, 52 (2003), 605. doi: 10.1080/02331930310001611420.

[1]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[2]

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

[3]

Chunrong Chen, T. C. Edwin Cheng, Shengji Li, Xiaoqi Yang. Nonlinear augmented Lagrangian for nonconvex multiobjective optimization. Journal of Industrial & Management Optimization, 2011, 7 (1) : 157-174. doi: 10.3934/jimo.2011.7.157

[4]

Giancarlo Bigi. Componentwise versus global approaches to nonsmooth multiobjective optimization. Journal of Industrial & Management Optimization, 2005, 1 (1) : 21-32. doi: 10.3934/jimo.2005.1.21

[5]

Sebastian Albrecht, Marion Leibold, Michael Ulbrich. A bilevel optimization approach to obtain optimal cost functions for human arm movements. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 105-127. doi: 10.3934/naco.2012.2.105

[6]

Dan Li, Li-Ping Pang, Fang-Fang Guo, Zun-Quan Xia. An alternating linearization method with inexact data for bilevel nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2014, 10 (3) : 859-869. doi: 10.3934/jimo.2014.10.859

[7]

Michael Hintermüller, Tao Wu. Bilevel optimization for calibrating point spread functions in blind deconvolution. Inverse Problems & Imaging, 2015, 9 (4) : 1139-1169. doi: 10.3934/ipi.2015.9.1139

[8]

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

[9]

Vadim Azhmyakov. An approach to controlled mechanical systems based on the multiobjective optimization technique. Journal of Industrial & Management Optimization, 2008, 4 (4) : 697-712. doi: 10.3934/jimo.2008.4.697

[10]

Truong Q. Bao, Boris S. Mordukhovich. Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1069-1096. doi: 10.3934/dcds.2011.31.1069

[11]

Chunyang Zhang, Shugong Zhang, Qinghuai Liu. Homotopy method for a class of multiobjective optimization problems with equilibrium constraints. Journal of Industrial & Management Optimization, 2017, 13 (1) : 81-92. doi: 10.3934/jimo.2016005

[12]

Hong-Zhi Wei, Chun-Rong Chen. Three concepts of robust efficiency for uncertain multiobjective optimization problems via set order relations. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2018066

[13]

Radu Ioan Boţ, Anca Grad, Gert Wanka. Sequential characterization of solutions in convex composite programming and applications to vector optimization. Journal of Industrial & Management Optimization, 2008, 4 (4) : 767-782. doi: 10.3934/jimo.2008.4.767

[14]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[15]

Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial & Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697

[16]

Oliver Jenkinson. Ergodic Optimization. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197

[17]

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

[18]

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

[19]

Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385

[20]

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 & Optimization, 2012, 2 (1) : 167-185. doi: 10.3934/naco.2012.2.167

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]