International Journal of Nonlinear Analysis and Applications

Some Pareto optimality results for nonsmooth multiobjective optimization problems with equilibrium constraints

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran

2 Department of Mathematics, Faculty of Sciences, University of Qom, Qom, Iran

3 Department of Mathematics and Computer Science, University of Qom, Qom, Iran

Abstract

In this paper, we study the nonsmooth multiobjective optimization problems with equilibrium constraints (MOMPEC). First, we extend the Guignard constraint qualification for MOMPEC, and then more constraint qualifications are developed. Also, the relationships between them are investigated. Moreover, we introduce the notion of primal Pareto stationarity and some dual Pareto stationarity concepts for a feasible point of MOMPEC. Some necessary optimality conditions are derived for any Pareto optimality solution of MOMPEC under weak assumptions. Indeed, we just need the objective functions to be locally Lipschitz. Further, we indicate our defined Pareto stationarity concepts are also sufficient conditions under the generalized convexity requirements.

Keywords

[1] A.A. Ardali, N. Movahedian and S. Nobakhtian, Optimality conditions for nonsmooth mathematical programs
with equilibrium constraints, using convexificators, Optim. 65 (2014), 67–85.
[2] A.A. Ardali, N. Movahedian and S. Nobakhtian, Optimality conditions for nonsmooth equilibrium problems via
Hadamard directional derivative, Set-Valued Var. Anal. 24 (2016), 483–497.[3] J.M. Borwein and A.S. Lewis, Convex analysis and nonlinear optimization: Theory and examples, Springer-Verlag,
New York, 2010.
[4] V.F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, J. Glob. Optim. 10 (1997),
305–326.
[5] V.F. Demyanov, Convexification and concavification of a positively homogenous function by the same family of
linear functions, Report 3, 208, 802. Universita di Pisa, 1994.
[6] J. Dutta and S. Chandra, Convexificators, generalized convexity and vector optimization, Optim. 53 (2004), 77–94.
[7] M.L. Flegel, C. Kanzow, A Fritz John approach to first order optimality conditions for mathematical programs
with equilibrium constraints, Optim. 52 (2003), 277–286.
[8] ML. Flegel, C. Kanzow, On the Guignard constraint qualification for mathematical programs with equilibrium
constraints, Optim. 54 (2005), 517–534.
[9] V. Jeyakumar and D.T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators J. Optim.
Theory Appl. 101 (1999), 599–621.
[10] G.H. Lin, D.L. Zhang and Y.C. Liang, Stochastic multiobjective problems with complementarity constraints and
applications in healthcare management, Eur. J. Oper. Res. 226 (2013), 461–470.
[11] X.F. Li and J.Z. Zhang, Stronger Kuhn-Tucker type conditions in nonsmooth multiobjective optimization: Locally
Lipschitz case, J. Optim. Theory Appl. 127 (2005), 367–388.
[12] Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical programs with equilibrium constraints, Cambridge University
Press, Cambridge, 1996.
[13] N. Movahedian and S. Nobakhtian, Necessary and sufficient conditions for nonsmooth mathematical programs
with equilibrium constraints, Nonlinear Anal. 72 (2010), 2694–2705.
[14] B.S. Mordukhovich, Multiobjective optimization problems with equilibrium constraints, Math. Program. 117
(2009), 331–354.
[15] B.S. Mordukhovich,Variational analysis and generalized differentiation, 330, Springer-Verlag, Berlin, 2006.
[16] T. Maeda, Constraint qualifications in multiobjective optimization problems: Differentiable case, J. Optim. Theory
Appl. 80 (1994), 483–500.
[17] JV. Outrata, M. Ko´cvara and J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints,
Nonconvex Optim. Appl. 28. Kluwer Academic Publishers, Dordrecht, 1998.
[18] X. Wang and V. Jeykumar, A sharp lagrangian multiplier rule for nonsmooth mathematical programming problems
involving equality constraints, SIAM J. Optim. 10 (2000), 1136–1148.
[19] J.J. Ye, Necessary optimality conditions for multiobjective bilevel programs Math. Oper. Res. 36 (2011), 165–184.
[20] J.J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J.
Math. Anal. Appl. 307 (2005), 350–369.
[21] J.J. Ye and Q.J. Zhu, Multiobjective optimization problem with variational inequality constraints, Math. Program.
96 (2003), 139–160.
[22] P. Zhang, J. Zhang, GH. Lin and X. Yang, Some kind of Pareto stationarity for multiobjective problems with
equilibrium constraints, Optim. 68 (2019), 1245–1260.
[23] P. Zhang, J. Zhang and G.H. Lin, Constraint qualifications and proper Pareto optimality conditions for multiobjective problems with equilibrium constraints J. Optim. Theory Appl. 176 (2018), 763–782.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 2185-2196
  • Receive Date: 05 November 2021
  • Revise Date: 05 January 2022
  • Accept Date: 21 January 2022