January  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 and 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), 179-203. 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), 577-604. 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), 2199-2227. doi: 10.1137/090766413.

[4]

M. Kočvara, M. Kružìk and J. Outrata, On the control of an evolutionary equilibrium in micromagnetics, in "Optimization with Multivalued Mappings" (eds. S. Dempe and V. Kalashnikov), Springer, New York, (2006), 143-168. doi: 10.1007/0-387-34221-4_8.

[5]

M. Kočvara and J. Outrata, On the modeling and control of delamination processes, in "Control and Boundary Analysis" (eds. J. Cagnol and J. -P. Yolesion), Marcel Dekker, (2004), 171-190.

[6]

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

[7]

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

[8]

B. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech., 40 (1976), 960-969. doi: 10.1016/0021-8928(76)90136-2.

[9]

R. Rockafellar and R. Wets, "Variational Analysis," Springer, Berlin, 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 "Global Engineering Management: Emerging Trends in the Asia Pacific," Proceedings of 1995 IEEE Annual International Publication, (1995), 370-377.

[11]

R. Zhang, Multistage bilevel programming problems, Optimization, 52 (2003), 605-616. 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), 179-203. 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), 577-604. 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), 2199-2227. doi: 10.1137/090766413.

[4]

M. Kočvara, M. Kružìk and J. Outrata, On the control of an evolutionary equilibrium in micromagnetics, in "Optimization with Multivalued Mappings" (eds. S. Dempe and V. Kalashnikov), Springer, New York, (2006), 143-168. doi: 10.1007/0-387-34221-4_8.

[5]

M. Kočvara and J. Outrata, On the modeling and control of delamination processes, in "Control and Boundary Analysis" (eds. J. Cagnol and J. -P. Yolesion), Marcel Dekker, (2004), 171-190.

[6]

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

[7]

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

[8]

B. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech., 40 (1976), 960-969. doi: 10.1016/0021-8928(76)90136-2.

[9]

R. Rockafellar and R. Wets, "Variational Analysis," Springer, Berlin, 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 "Global Engineering Management: Emerging Trends in the Asia Pacific," Proceedings of 1995 IEEE Annual International Publication, (1995), 370-377.

[11]

R. Zhang, Multistage bilevel programming problems, Optimization, 52 (2003), 605-616. doi: 10.1080/02331930310001611420.

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