Optimality Conditions Using Convexifactors for a Multiobjective Fractional Bilevel Programming Problem

Authors

  • Bhawna Kohli Delhi university

DOI:

https://doi.org/10.5556/j.tkjm.54.2023.3830

Keywords:

Keywords Multiobjective fractional bilevel programming problem, Convexifactors, Value function, Constraint qualifications.

Abstract

In this paper, a multiobjective fractional bilevel programming problem is considered and optimality conditions using the concept of convexifactors are established for it. For this purpose, a suitable constraint qualification in terms of convexifactors is introduced for the problem. Further in the paper, notions of asymptotic pseudoconvexity, asymptotic quasiconvexity in terms of convexifactors are given and using them sufficient optimality conditions are derived.

References

J. F. Bard, Optimality conditions for the bilevel programming problem, Naval Research Logistics Quarterly 31 (1984), 13-26.

J. F. Bard, Some properties of the bilevel programming problem, Journal of Optimization Theory and Applications 68 (1991), 371-378.

J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, Kluwer Academic Publications, The Netherlands (1998).

C. R. Bector, S. Chandra, J. Dutta, Principles of Optimization Theory, Narosa Publications (2005).

H. Bonnel, Optimality conditions for the semivectorial bilevel optimization problem, {Pacific Journal of Optimization 2(3) (2006), 447-467.

H. Bonnel, J. Morgan, Semivectorial bilevel optimization problem; penalty approach, {Journal of Optimization Theory and Applications 131(3) (2006), 365-382.

H. Bonnel, J. Morgan, Optimality conditions for semivectorial bilevel convex optimal control problem, In: H. Bauschke, M. Thera (eds.), {Computational and Analytical Mathematics, Springer Proceeding in Mathematics 50 (2013), 45-78.

K. Bouibed, S. Hachem, M. S. Radjef, Global efficiency for multiobjective bilevel programming problems under generalized invexity, Journal of Applied Mathematics and Computing, 53, no. 1-2, 507-530 (2017)

H. I. Calvete, C. Gale, The bilevel linear/ linear fractional programming problem, European Journal of Operational Research 114 (1) (1999), 188-197.

H. I. Calvete, C.. Gale, A note on `Bi-Level linear fractional programming problem', European Journal of Operational Research 152(1) (2004(a)), 296-299.

H. I. Calvete, C. Gale, Optimality conditions for the linear fractional/ quadratic bilevel problem, Monografias del Seminario Matematico Garcia de Galdeano 31 (2004), 285-294.

S. Dempe, A necessary and sufficient optimality condition for bilevel programming problem, Optimization 25 (1992), 341-354.

S. Dempe, First order necessary optimality conditions for general bilevel programming problems, Journal of Optimization Theory and Applications 95 (1997), 735-739.

S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers (2002).

S. Dempe, J. Dutta, B. S. Mordukhovich, New necessary optimality conditions in optimistic bilevel programming, Optimization 56 (2007), 577-604.

V. F. Demyanov, Convexification and concavification of positively homogeneous function by the same family of linear functions, Report 3.208,802, Universita di pisa (1994).

J. Dutta, S. Chandra, Convexifactors, generalized convexity and optimality conditions, Journal of Optimization Theory and Applications 113 (2002), 41-65.

J. Dutta, S. Chandra, Convexifactors, generalized convexity and vector optimization, Optimization 53 (2004), 77-94.

N. Gadhi, S. Dempe, Necessary optimality conditions and a new approach to multiobjective bilevel optimization problems, Journal of Optimization Theory and Applications 155(1) (2014), 100-114.

V. Jeyakumar, D. T. Luc, Nonsmooth calculus, maximality and monotonicity of convexificators, Journal of Optimization Theory and Applications 101 (1999), 599-621.

A. Jourani, Constraint qualifications and lagrange multipliers in nondifferentiable programming problems, Journal of Optimization Theory and Applications 81(3) (1994), 533-548.

A. Kabgani, M. Soleimani-damaneh, Characterizations of (weakly/properly/roboust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators, Optimization 67 (2) (2018), 217-235.

A. Kabgani, M. Soleimani-damaneh, The Relationships between convexificators and Greensberg-Pierskalla subdifferentials for quasiconvex functions, Numerical Functional Analysis and Optimization 38 (12) (2017), 1548-1563.

A. Kabgani, M. Soleimani-damaneh, M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexifactors, Mathematical Methods of Operations Research 86 (2017), 103-121.

B. Kohli, Optimality conditions for optimistic bilevel programming problem using convexifactors, Journal of Optimization Theory and Applications 152 (3) (2012), 632-651.

B. Kohli, Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints, Rairo Operations Research 53 (2019), 1617-1632.

X. F. Li, J. Z. Zhang, Necessary optimality conditions in terms of convexificators in lipschitz optimization, Journal of Optimization Theory and Applications 131 (2006), 429-452.

D. V. Luu, Optimality condition for local efficient solutions of vector equilibrium problems via convexificators and applications, Journal of Optimization Theory and Applications 171 (2) (2016), 643-665.

D.V. Luu, T.T. Mai, Efficiency conditions for multiobjective bilevel programming problems via convexificators, J. Nonlinear Var. Anal. 4(3) (2020), 399-414.

D. V. Luu, P. T. Linh, Optimality and duality for nonsmooth multiobjective fractional problems using convexificators, J. Nonlinear Funct. Anal. (2021), Article ID 1.

J. V. Outrata, On necessary optimality conditions for stackelberg problems, Journal of Optimization Theory and Applications 76 (1993), 306-320.

S. K. Suneja, B. Kohli, Optimality and duality results for bilevel programming problem using convexifactors, Journal of Optimization Theory and Applications 150 (2011), 1-19.

S. K. Suneja, B. Kohli, Generalized nonsmooth cone convexity in terms of convexifactors in vector optimization, Opsearch 50 (1) (2013), 89-105.

S. K. Suneja, B. Kohli, Duality for multiobjective fractional programming problem using convexifactors, Mathematical Sciences 7:6 (2013), 8 pages.

L. T. Tung, Karush-Kuhn-Tucker optimality conditions for nonsmooth multiobjective semidefinite and semi-infinite programming, J. Appl. Numer. Optim. 1(1) (2019), 63-75.

J. J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Research 36 (1) (2011), 165-184.

J. J. Ye, Nondifferentiable multiplier rules for optimization and bilevel optimization problems, Siam Journal of Optimization 15 (2004), 252-274.

J. J. Ye, Constraint qualifications and KKT conditions for bilevel programming problems, Mathematics of Operations Research 31 (2006), 811-824.

Downloads

Published

2021-11-20

How to Cite

Kohli, B. (2021). Optimality Conditions Using Convexifactors for a Multiobjective Fractional Bilevel Programming Problem . Tamkang Journal of Mathematics, 54. https://doi.org/10.5556/j.tkjm.54.2023.3830

Issue

Section

Papers