Solution Concepts and New Optimality Conditions in Bilevel Multiobjective Programming

Affiliation(s)

Department of Mathematics, Faculty of Science, University of Yaoundé I, Yaoundé, Cameroon.

Department of Computer Sciences, Faculty of Science, University of Yaoundé I, Yaoundé, Cameroon.

Regional Bureau for Education in Africa, Pole de Dakar, UNESCO, Dakar, Senegal.

Department of Mathematics, Faculty of Science, University of Yaoundé I, Yaoundé, Cameroon.

Department of Computer Sciences, Faculty of Science, University of Yaoundé I, Yaoundé, Cameroon.

Regional Bureau for Education in Africa, Pole de Dakar, UNESCO, Dakar, Senegal.

ABSTRACT

In this paper, new sufficient optimality theorems for a solution of a differentiable bilevel multiobjective optimization problem (BMOP) are established. We start with a discussion on solution concepts in bilevel multiobjective programming; a theorem giving necessary and sufficient conditions for a decision vector to be called a solution of the BMOP and a proposition giving the relations between four types of solutions of a BMOP are presented and proved. Then, under the pseudoconvexity assumptions on the upper and lower level objective functions and the quasiconvexity assumptions on the constraints functions, we establish and prove two new sufficient optimality theorems for a solution of a general BMOP with coupled upper level constraints. Two corollary of these theorems, in the case where the upper and lower level objectives and constraints functions are convex are presented.

In this paper, new sufficient optimality theorems for a solution of a differentiable bilevel multiobjective optimization problem (BMOP) are established. We start with a discussion on solution concepts in bilevel multiobjective programming; a theorem giving necessary and sufficient conditions for a decision vector to be called a solution of the BMOP and a proposition giving the relations between four types of solutions of a BMOP are presented and proved. Then, under the pseudoconvexity assumptions on the upper and lower level objective functions and the quasiconvexity assumptions on the constraints functions, we establish and prove two new sufficient optimality theorems for a solution of a general BMOP with coupled upper level constraints. Two corollary of these theorems, in the case where the upper and lower level objectives and constraints functions are convex are presented.

Cite this paper

F. Dedzo, L. Fotso and C. Pieume, "Solution Concepts and New Optimality Conditions in Bilevel Multiobjective Programming,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1395-1402. doi: 10.4236/am.2012.330196.

F. Dedzo, L. Fotso and C. Pieume, "Solution Concepts and New Optimality Conditions in Bilevel Multiobjective Programming,"

References

[1] H. Von Stackelberg, “The Theory of Market Economy,” Oxford University Press, Oxford, 1952.

[2] P. Heiskanen, “Decentralized Method for Computing Pareto Solutions in Multiparty Negotiations,” European Journal of Operational Research, Vol. 117, No. 3, 1999, 578-590. doi:10.1016/S0377-2217(98)00276-8

[3] S. Dempe, “Foundations of Bilevel Programming,” Kluwer Academic Publisher, 2002.

[4] S. Dempe, “Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints,” Optimization, Vol. 52, No. 3, 2003, pp. 333-359. doi:10.1080/0233193031000149894

[5] L. N. Vicente and P. H. Calamai, “Bilevel and Multilevel Programming: A Bibliography Review,” Journal of Global Optimization, Vol. 5, No. 3, 1994, pp. 291-306. doi:10.1007/BF01096458

[6] B. Colson, P. Marcotte and G. Savard, “An Overview of Bilevel Optimization,” Annals of Operational Research, Vol. 153, No. 1, 2007, pp. 235-256.

[7] A. Dell’Aere, “Numerical Methods for the Solution of Bi-Level Multi-Objective Optimization Problems,” Ph.D. Thesis, University of Paderborn, Paderborn, 2008.

[8] K. Deb and A. Sinha, “Solving Bilevel Multi-Objective Optimization Problems Using Evolutionary Algorithms,” Technical Report KanGAL Report No. 2008005, Indian Institute of Technology, Kanpur, 2008.

[9] C. O. Pieume, L. P. Fotso and P. Siarry, “Solving Bilevel Programming Problems with Multicriteria Optimization Techniques,” OPSEARCH, Vol. 46, No. 2, 2009, pp. 169-183.

[10] G. Eichfelder, “Multiobjective Bilevel Optimization,” Mathematical Programming, Vol. 123, No. 2, 2008, pp. 419-449. doi:10.1007/s10107-008-0259-0

[11] G. Eichfelder, “Solving Nonlinear Multiobjective Bilevel Optimization Problems with Coupled Upper Level Constraints,” Technical Report Preprint No. 320, PreprintSeries of the Institute of Applied Mathematics, University Erlangen-N?ijrnberg, Erlangen and Nuremberg, 2007.

[12] C. O. Pieume, P. Marcotte, L. P. Fotso and P. Siarry, “Solving Bilevel Linear Multiobjective Programming Problems,” American Journal of Operations Research, Vol. 1 No. 4, 2011, pp. 214-219.

[13] A. A. K. Majumdar, “Optimality Conditions in Differentiable Multiobjective Programming,” Journal of Optimization Theory and Applications, Vol. 92, No. 2, 1997, pp. 419-427. doi:10.1023/A:1022667432420

[14] D. S. Kim, G. M. Lee, B. S. Lee and S. J. Cho, “Counterexample and Optimality Conditions in Differentiable Multiobjective Programming,” Journal of Optimization Theory and Applications, Vol. 109, No. 1, 2001, pp. 187-192. doi:10.1023/A:1017570023018

[15] J. J. Ye, “Necessary Optimality Conditions for Multiobjective Bilevel Programs,” Journal of the Institute for Operations Research and the Management Sciences, Vol. 36, No. 1, 2010, pp. 165-184.

[16] S. Dempe, N. Gadhi and A. B. Zemkoho, “New Optimality Conditions for the Semivectorial Bilevel Optimization Problem,” 2011.

[17] P.-Y. Nie, “A Note on Bilevel Optimization Problems,” International Journal of Applied Mathematical Sciences, Vol. 2, No. 1, 2005, pp. 31-38.

[18] O. L. Mangasarian, “Nonlinear Programming,” Mc GrawHill Book Company, New York, 1969.

[1] H. Von Stackelberg, “The Theory of Market Economy,” Oxford University Press, Oxford, 1952.

[2] P. Heiskanen, “Decentralized Method for Computing Pareto Solutions in Multiparty Negotiations,” European Journal of Operational Research, Vol. 117, No. 3, 1999, 578-590. doi:10.1016/S0377-2217(98)00276-8

[3] S. Dempe, “Foundations of Bilevel Programming,” Kluwer Academic Publisher, 2002.

[4] S. Dempe, “Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints,” Optimization, Vol. 52, No. 3, 2003, pp. 333-359. doi:10.1080/0233193031000149894

[5] L. N. Vicente and P. H. Calamai, “Bilevel and Multilevel Programming: A Bibliography Review,” Journal of Global Optimization, Vol. 5, No. 3, 1994, pp. 291-306. doi:10.1007/BF01096458

[6] B. Colson, P. Marcotte and G. Savard, “An Overview of Bilevel Optimization,” Annals of Operational Research, Vol. 153, No. 1, 2007, pp. 235-256.

[7] A. Dell’Aere, “Numerical Methods for the Solution of Bi-Level Multi-Objective Optimization Problems,” Ph.D. Thesis, University of Paderborn, Paderborn, 2008.

[8] K. Deb and A. Sinha, “Solving Bilevel Multi-Objective Optimization Problems Using Evolutionary Algorithms,” Technical Report KanGAL Report No. 2008005, Indian Institute of Technology, Kanpur, 2008.

[9] C. O. Pieume, L. P. Fotso and P. Siarry, “Solving Bilevel Programming Problems with Multicriteria Optimization Techniques,” OPSEARCH, Vol. 46, No. 2, 2009, pp. 169-183.

[10] G. Eichfelder, “Multiobjective Bilevel Optimization,” Mathematical Programming, Vol. 123, No. 2, 2008, pp. 419-449. doi:10.1007/s10107-008-0259-0

[11] G. Eichfelder, “Solving Nonlinear Multiobjective Bilevel Optimization Problems with Coupled Upper Level Constraints,” Technical Report Preprint No. 320, PreprintSeries of the Institute of Applied Mathematics, University Erlangen-N?ijrnberg, Erlangen and Nuremberg, 2007.

[12] C. O. Pieume, P. Marcotte, L. P. Fotso and P. Siarry, “Solving Bilevel Linear Multiobjective Programming Problems,” American Journal of Operations Research, Vol. 1 No. 4, 2011, pp. 214-219.

[13] A. A. K. Majumdar, “Optimality Conditions in Differentiable Multiobjective Programming,” Journal of Optimization Theory and Applications, Vol. 92, No. 2, 1997, pp. 419-427. doi:10.1023/A:1022667432420

[14] D. S. Kim, G. M. Lee, B. S. Lee and S. J. Cho, “Counterexample and Optimality Conditions in Differentiable Multiobjective Programming,” Journal of Optimization Theory and Applications, Vol. 109, No. 1, 2001, pp. 187-192. doi:10.1023/A:1017570023018

[15] J. J. Ye, “Necessary Optimality Conditions for Multiobjective Bilevel Programs,” Journal of the Institute for Operations Research and the Management Sciences, Vol. 36, No. 1, 2010, pp. 165-184.

[16] S. Dempe, N. Gadhi and A. B. Zemkoho, “New Optimality Conditions for the Semivectorial Bilevel Optimization Problem,” 2011.

[17] P.-Y. Nie, “A Note on Bilevel Optimization Problems,” International Journal of Applied Mathematical Sciences, Vol. 2, No. 1, 2005, pp. 31-38.

[18] O. L. Mangasarian, “Nonlinear Programming,” Mc GrawHill Book Company, New York, 1969.