The Improvement of Optimality Test over Possible Reaction Set in Bilevel Linear Optimization with Ambiguous Objective Function of the Follower

Puchit Sariddichainunta; Masahiro Inuiguchi

doi:10.20965/jaciii.2015.p0645

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JACIII Vol.19 No.5 pp. 645-654

doi: 10.20965/jaciii.2015.p0645

(2015)

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The Improvement of Optimality Test over Possible Reaction Set in Bilevel Linear Optimization with Ambiguous Objective Function of the Follower

Puchit Sariddichainunta and Masahiro Inuiguchi

Graduate School of Engineering Science, Osaka University
1-3 Kanemachiyama, Toyonaka, Osaka 560-8531, Japan

Received:

January 31, 2015

Accepted:

June 16, 2015

Published:

September 20, 2015

Keywords:

bilevel linear programming, degenerate solution, maximin optimization

Abstract

Verifying a rational response is the most crucial step in searching for an optimal solution in bilevel linear programming. Such verification is even difficult in a model with ambiguous objective function of the follower who reacts rationally to a leader’s decision. In our model, we assume that the ambiguous coefficient vector of follower lies in a convex polytope and we formulate bilevel linear programming with the ambiguous objective function of the follower as a special three-level programming problem. We use the k-th best method that sequentially enumerates a solution and examine whether it is the best of all possible reactions. The optimality test process over possible reactions in lower-level problems usually encounters degenerate bases that become obstacles to verifying the optimality of an enumerated solution efficiently. To accelerate optimality verification, we propose search strategies and the evaluation of basic possible reactions adjacent to a degenerate basic solution. We introduce these methods in both local and global optimality testing, confirming the effectiveness of our proposed methods in numerical experiments.

Cite this article as:

P. Sariddichainunta and M. Inuiguchi, “The Improvement of Optimality Test over Possible Reaction Set in Bilevel Linear Optimization with Ambiguous Objective Function of the Follower,” J. Adv. Comput. Intell. Intell. Inform., Vol.19 No.5, pp. 645-654, 2015.

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References

[1] J. F. Bard, “Practical Bilevel Optimization: Algorithms and Applications,” Dordrecht: Kluwer Academic Publishers, 1998.
[2] S. Dempe, “Foundations of Bilevel Programming,” Dordrecht: Kluwer Academic Publishers, 2002.
[3] W. Bialas and M. Karwan, “On two-level optimization,” IEEE Trans. on Automatic Control, Vol.27, No.1, pp. 211-214, Feb. 1982.
[4] S. Dempe, “Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints,” Optimization: A J. of Mathematical Programming and Operations Research, Vol.52, No.3, pp. 333-359, 2003.
[5] S. A. Abass, “An interval number programming approach for bilevel linear programming problem,” Int. J. of Management Science and Engineering Management, Vol.5, No.6, pp. 461-464, 2010.
[6] J. Z. Wang and G. Du, “Research on the method for interval linear bi-level programming based on partial order on intervals,” Proc. of 8th Int. Conf. on Fuzzy Systems and Knowledge Discovery IEEE, pp. 682-686, 2001.
[7] H. I. Calvete and C. Galée, H. I.“Linear bilevel programming with interval coefficients,” J. of Computational and Applied Mathematics, Vol.236, No.15, pp. 3751-3762, 2012.
[8] A. Ren and Y. Wang, “A cutting plane method for bilevel linear programming with interval coefficients,” Annals of Operations Research, Vol.233, No.1, pp. 355-378, 2014.
[9] M. Inuiguchi, P. Sariddichainunta, and Y. Kawase, “Bilevel linear programming with ambiguous objective function of the follower: Formulation and Algorithm,” Proc. of the 8^th Int. Conf. on Nonlinear Analysis and Convex Analysis, pp. 207-217, 2013.
[10] M. Inuiguchi and Y. Kume, “Minimax Regret in Linear Programming Problems with an Interval Objective Function,” in: G. H. Tzeng, H. F. Wang, U. P. Wen and P. L. Yu (eds.), Multiple Criteria Decision Making, Springer, NY, pp. 65-74, 1994.
[11] R. E. Steuer, “Multiple Criteria Optimization: Theory, Computation, and Application,” New York: John Wiley and Sons, 1986.
[12] M. Inuiguchi and M. Sakawa, “Possible and necessary optimality tests in possibilistic linear programming problems,” Fuzzy Sets and Systems, Vol.67, No. 1, pp. 29-46, Oct. 1994.
[13] M. Inuiguchi and T. Tanino, “Enumeration of all possibly optimal vertices with possible optimality degrees in linear programming problems with a possibilistic objective function,” Fuzzy Optimization and Decision Making, Vol.3, No.4, pp. 311-326, Dec. 2004.
[14] M. E. Muller, “A note on a method for generating points uniformly on n-dimensional spheres,” Communications of the ACM, Vol.2, No.4, pp. 19-20, 1959.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] J. F. Bard, “Practical Bilevel Optimization: Algorithms and Applications,” Dordrecht: Kluwer Academic Publishers, 1998.

[2] [2] S. Dempe, “Foundations of Bilevel Programming,” Dordrecht: Kluwer Academic Publishers, 2002.

[3] [3] W. Bialas and M. Karwan, “On two-level optimization,” IEEE Trans. on Automatic Control, Vol.27, No.1, pp. 211-214, Feb. 1982.

[4] [4] S. Dempe, “Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints,” Optimization: A J. of Mathematical Programming and Operations Research, Vol.52, No.3, pp. 333-359, 2003.

[5] [5] S. A. Abass, “An interval number programming approach for bilevel linear programming problem,” Int. J. of Management Science and Engineering Management, Vol.5, No.6, pp. 461-464, 2010.

[6] [6] J. Z. Wang and G. Du, “Research on the method for interval linear bi-level programming based on partial order on intervals,” Proc. of 8th Int. Conf. on Fuzzy Systems and Knowledge Discovery IEEE, pp. 682-686, 2001.

[7] [7] H. I. Calvete and C. Galée, H. I.“Linear bilevel programming with interval coefficients,” J. of Computational and Applied Mathematics, Vol.236, No.15, pp. 3751-3762, 2012.

[8] [8] A. Ren and Y. Wang, “A cutting plane method for bilevel linear programming with interval coefficients,” Annals of Operations Research, Vol.233, No.1, pp. 355-378, 2014.

[9] [9] M. Inuiguchi, P. Sariddichainunta, and Y. Kawase, “Bilevel linear programming with ambiguous objective function of the follower: Formulation and Algorithm,” Proc. of the 8^th Int. Conf. on Nonlinear Analysis and Convex Analysis, pp. 207-217, 2013.

[10] [10] M. Inuiguchi and Y. Kume, “Minimax Regret in Linear Programming Problems with an Interval Objective Function,” in: G. H. Tzeng, H. F. Wang, U. P. Wen and P. L. Yu (eds.), Multiple Criteria Decision Making, Springer, NY, pp. 65-74, 1994.

[11] [11] R. E. Steuer, “Multiple Criteria Optimization: Theory, Computation, and Application,” New York: John Wiley and Sons, 1986.

[12] [12] M. Inuiguchi and M. Sakawa, “Possible and necessary optimality tests in possibilistic linear programming problems,” Fuzzy Sets and Systems, Vol.67, No. 1, pp. 29-46, Oct. 1994.

[13] [13] M. Inuiguchi and T. Tanino, “Enumeration of all possibly optimal vertices with possible optimality degrees in linear programming problems with a possibilistic objective function,” Fuzzy Optimization and Decision Making, Vol.3, No.4, pp. 311-326, Dec. 2004.

[14] [14] M. E. Muller, “A note on a method for generating points uniformly on n-dimensional spheres,” Communications of the ACM, Vol.2, No.4, pp. 19-20, 1959.