Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints
RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 5, pp. 1617-1632.

The main aim of this paper is to develop necessary Optimality conditions using Convexifactors for mathematical programs with equilibrium constraints (MPEC). For this purpose a nonsmooth version of the standard Guignard constraint qualification (GCQ) and strong stationarity are introduced in terms of convexifactors for MPEC. It is shown that Strong stationarity is the first order necessary optimality condition under nonsmooth version of the standard GCQ. Finally, notions of asymptotic pseudoconvexity and asymptotic quasiconvexity are used to establish the sufficient optimality conditions for MPEC.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2018084
Classification : 90C30, 90C46, 49J52
Mots-clés : Mathematical programs with equilibrium constraints, Convexifactors, Guignard constraint qualification, Strong stationarity, Optimality conditions
Kohli, Bhawna 1

1 
@article{RO_2019__53_5_1617_0,
     author = {Kohli, Bhawna},
     title = {Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {1617--1632},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {5},
     year = {2019},
     doi = {10.1051/ro/2018084},
     zbl = {1431.90151},
     mrnumber = {4016525},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ro/2018084/}
}
TY  - JOUR
AU  - Kohli, Bhawna
TI  - Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2019
SP  - 1617
EP  - 1632
VL  - 53
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro/2018084/
DO  - 10.1051/ro/2018084
LA  - en
ID  - RO_2019__53_5_1617_0
ER  - 
%0 Journal Article
%A Kohli, Bhawna
%T Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2019
%P 1617-1632
%V 53
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro/2018084/
%R 10.1051/ro/2018084
%G en
%F RO_2019__53_5_1617_0
Kohli, Bhawna. Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 5, pp. 1617-1632. doi : 10.1051/ro/2018084. http://archive.numdam.org/articles/10.1051/ro/2018084/

[1] I. Ahmad, K. Kummari, V. Singh and A. Jayswal, Optimality and duality for nonsmooth minimax programming problems using convexifactors. Filomat 31 (2017) 4555–4570. | DOI | MR | Zbl

[2] Q.H. Ansari, C.S. Lalitha and M. Mehta, Generalized convexity, nonsmooth variational inequalities and nonsmooth optimization. CRC Press, New York, NY (2014). | MR | Zbl

[3] A.A. Ardali, N. Movahedian and S. Nobakhtian, Optimality conditions for nonsmooth mathematical programs with equilibrium constraints, using convexifactors. Optimization 65 (2016) 67–85. | DOI | MR | Zbl

[4] H. Babahadda and N. Gadhi, Necessary optimality conditions for bilevel optimization problems using convexificators. J. Global Optim. 34 (2006) 535–549. | DOI | MR | Zbl

[5] T.B. Baumrucker, G.J. Renfro and T.L. Biegler, MPEC problem formulations and solution strategies with chemical engineering applications. Comput. Chem. Eng. 32 (2008) 2903–2913. | DOI

[6] 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).

[7] J. Dutta and S. Chandra, Convexifactors, generalized convexity and optimality conditions. J. Optim. Theory App. 113 (2002) 41–65. | DOI | MR | Zbl

[8] J. Dutta and S. Chandra, Convexifactors, generalized convexity and vector optimization. Optimization 53 (2004) 77–94. | DOI | MR | Zbl

[9] L.M. Flegel and C. Kanzow, A Fritz John approach to first order optimality conditions for mathematical programs with equilibrium constraints. Optimization 52 (2003) 277–286. | DOI | MR | Zbl

[10] L.M. Flegel and C. Kanzow, On the guignard constraint qualification for mathematical programs with equilibrium constraints. Optimization 54 (2005) 517–534. | DOI | MR | Zbl

[11] L.M. Flegel and C. Kanzow, On M-stationary points for mathematical programs with equilibrium constraints. J. Math. Anal. App. 310 (2005) 286–302. | DOI | MR | Zbl

[12] L.M. Flegel and C. Kanzow, Abadie-Type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory App. 124 (2005) 595–614. | DOI | MR | Zbl

[13] L.M. Flegel, C. Kanzow and V.J. Outrata, Optimality conditions for disjunctive programs with applications to mathematical programs with equilibrium constraints. Set-Valued Anal. 15 (2007) 139–162. | DOI | MR | Zbl

[14] L. Guo and G. Lin, Notes on some constraint qualifications for mathematical programs with equilibrium constraints. J. Optim. Theory App. 156 (2013) 600–616. | DOI | MR | Zbl

[15] R. Henrion, J. Outrata and T. Surroviec, A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints. Kybernetika 46 (2010) 423–434. | MR | Zbl

[16] A. Jayswal, K. Kummari and V. Singh, Duality for a class of nonsmooth multiobjective programming problems using convexifactors. Filomat 31 (2017) 489–498. | DOI | MR | Zbl

[17] A. Jayswal, I. Stancu-Minasian and J. Banerjee, Optimality conditions and duality for interval-valued optimization problems using convexifactors. Rendiconti del Circolo 65 (2016) 17–32. | MR | Zbl

[18] V. Jeyakumar and D.T. Luc, Nonsmooth calculus, maximality and monotonicity of convexificators. J. Optim. Theory App. 101 (1999) 599–621. | DOI | MR | Zbl

[19] A. Kabgani and M. Soleimani-Damaneh, Characterizations of (weakly/properly/roboust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators. Optimization 67 (2018) 217–235. | DOI | MR | Zbl

[20] A. Kabgani and M. Soleimani-Damaneh, The relationships between convexifactors and Greensberg-Pierskalla subdifferentials for quasiconvex functions. Numer. Funct. Anal. Optim. 38 (2017) 1548–1563. | DOI | MR | Zbl

[21] A. Kabgani, M. Soleimani-Damaneh and M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexifactors. Math. Methods Oper. Res. 86 (2017) 103–121. | DOI | MR | Zbl

[22] C. Kanzow and A. Shwartz, Mathematical programs with equilibrium constraints: enhanced Fritz-John conditions, new constraint qualifications and improved exact penalty results. SIAM J. Optim. 20 (2010) 2730–2753. | DOI | MR | Zbl

[23] B. Kohli, Optimality conditions for optimistic bilevel programming problem using convexifactors. J. Optim. Theory App. 152 (2012) 632–651. | DOI | MR | Zbl

[24] X.F. Li and J.Z. Zhang, Necessary optimality conditions in terms of convexificators in Lipschitz optimization. J. Optim. Theory App. 131 (2006) 429–452. | DOI | MR | Zbl

[25] Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge (1996). | DOI | MR | Zbl

[26] D. Van Luu, Optimality conditions for local efficient solutions of vector equilibrium problems via convexificators and applications. J. Optim. Theory App. 171 (2016) 643–665. | DOI | MR | Zbl

[27] N. Movahedian, S. Nobakhtian, Constraint qualifications for nonsmooth mathematical programs with equilibrium constraints. Set-Valued Variational Anal. 17 (2009) 63–95. | DOI | MR | Zbl

[28] N. Movahedian and S. Nobakhtian, Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints. Nonlinear Anal. 72 (2010) 2694–2705. | DOI | MR | Zbl

[29] J. Outrata, Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Methods Oper. Res. 24 (1999) 627–644. | DOI | MR | Zbl

[30] J. Outrata, A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38 (2000) 1623–1638. | DOI | MR | Zbl

[31] J. Outrata, M. Koćvara and J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints: Theory, Applications and Numerical Results. Kluwer Academic, Boston (1998). | DOI | MR | Zbl

[32] J.S. Pang and M. Fukushima, Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints. Comput. Optim. App. 13 (1999) 111–136. | DOI | MR | Zbl

[33] U.A. Raghunathan, S.M. Diaz and T.L. Biegler, An MPEC formulation for dynamic optimization of distillation operations. Comput. Chem. Eng. 28 (2008) 2037–2052. | DOI

[34] H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity. Math. Oper. Res. 25 (2000) 1–22. | DOI | MR | Zbl

[35] S.K. Suneja and B. Kohli, Optimality and duality results for bilevel programming problem using convexifactors. J. Optim. Theory App. 150 (2011) 1–19. | DOI | MR | Zbl

[36] S.K. Suneja and B. Kohli, Generalized nonsmooth cone convexity in terms of convexifactors in vector optimization. Opsearch 50 (2013) 89–105. | DOI | MR | Zbl

[37] S.K. Suneja and B. Kohli, Duality for multiobjective fractional programming problem using convexifactors. Math. Sci. 7 (2013) 8. | DOI | MR | Zbl

[38] J.J. Ye, Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10 (2000) 943–962. | DOI | MR | Zbl

[39] J.J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. App. 307 (2005) 350–369. | DOI | MR | Zbl

[40] J.J. Ye and X.Y. Ye, Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22 (1997) 977–997. | DOI | MR | Zbl

Cité par Sources :