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November  2021, 17(6): 3209-3221. doi: 10.3934/jimo.2020114

Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem

Department of Mathematics, P.G.D.A.V. College, University of Delhi, Delhi-110065, India

Received  May 2019 Revised  March 2020 Published  November 2021 Early access  June 2020

The main aim of this paper is to establish sufficient optimality conditions using an upper estimate of Clarke subdifferential of value function and the concept of convexifactor for optimistic bilevel programming problems with convex and non-convex lower-level problems. For this purpose, the notions of asymptotic pseudoconvexity and asymptotic quasiconvexity are defined in terms of the convexifactors.

Citation: Bhawna Kohli. Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3209-3221. doi: 10.3934/jimo.2020114
References:
[1]

I. AhmadK. KummariV. Singh and A. Jayswal, Optimality and duality for nonsmooth minimax programming problems using convexifactors, Filomat, 31 (2017), 4555-4570.  doi: 10.2298/FIL1714555A.

[2]

J. F. Bard, Practical Bilevel Optimization. Algorithms and Applications, Nonconvex Optim. Appl., 30, Kluwer Acad. Publ., Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.

[3]

J. F. Bard, Optimality conditions for the bilevel programming problem, Naval Res. Logist. Quart., 31 (1984), 13-26.  doi: 10.1002/nav.3800310104.

[4]

J. F. Bard, Some properties of the bilevel programming problem, J. Optim. Theory Appl., 68 (1991), 371-378.  doi: 10.1007/BF00941574.

[5]

C. R. Bector, S. Chandra and J. Dutta, Principles of Optimization Theory, Narosa Publishing House, 2005.

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.

[7]

S. Dempe, Foundations of Bilevel Programming, Nonconvex Optim. Appl., 61, Kluwer Acad. Publ., Dordrecht, 2002. doi: 10.1007/b101970.

[8]

S. Dempe, A necessary and a sufficient optimality condition for bilevel programming problems, Optimization, 25 (1992), 341-354.  doi: 10.1080/02331939208843831.

[9]

S. Dempe, First-order necessary optimality conditions for general bilevel programming problems, J. Optim. Theory Appl., 95 (1997), 735-739.  doi: 10.1023/A:1022646611097.

[10]

S. DempeJ. Dutta and B. S. Mordukhovich, New necessary optimality conditions in optimistic bilevel programming, Optimization, 56 (2007), 577-604.  doi: 10.1080/02331930701617551.

[11]

V. F. Demyanov, Convexification and concavification of positively homogeneous function by the same family of linear functions, Report 3,208,802 from Universita di Pisa, 1994.

[12]

V. F. Demyanov and A. M. Rubinov, An introduction to quasidifferential calculus, in Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., 43, Kluwer Acad. Publ., Dordrecht, 2000, 1–31. doi: 10.1007/978-1-4757-3137-8_1.

[13]

J. Dutta and S. Chandra, Convexifactors, generalized convexity and optimality conditions, J. Optim. Theory Appl., 113 (2002), 41-64.  doi: 10.1023/A:1014853129484.

[14]

J. Dutta and S. Chandra, Convexifactors, generalized convexity and vector optimization, Optimization, 53 (2004), 77-94.  doi: 10.1080/02331930410001661505.

[15]

A. JayswalK. Kummari and V. Singh, Duality for a class of nonsmooth multiobjective programming problems using convexifactors, Filomat, 31 (2017), 489-498.  doi: 10.2298/FIL1702489J.

[16]

A. Jayswal, I. Stancu-Minasian and J. Banerjee, Optimality conditions and duality for interval-valued optimization problems using convexifactors, Rend. Circ. Mat. Palermo (2), 65 (2016), 17–32. doi: 10.1007/s12215-015-0215-9.

[17]

V. Jeyakumar and D. T. Luc, Nonsmooth calculus, maximality and monotonicity of convexificators, J. Optim. Theory Appl., 101 (1999), 599-621.  doi: 10.1023/A:1021790120780.

[18]

A. Kabgani and M. Soleimani-damaneh, Relationships between convexificators and Greensberg-Pierskalla subdifferentials for quasiconvex functions, Numer. Funct. Anal. Optim., 38 (2017), 1548-1563.  doi: 10.1080/01630563.2017.1349144.

[19]

A. KabganiM. 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: 10.1007/s00186-017-0584-2.

[20]

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: 10.1080/02331934.2017.1393675.

[21]

B. Kohli, Optimality conditions for optimistic bilevel programming problem using convexifactors, J. Optim. Theory Appl., 152 (2012), 632-651.  doi: 10.1007/s10957-011-9941-0.

[22]

B. Kohli, A note on the paper "Optimality conditions for optimistic bilevel programming problem using convexifactors", J. Optim. Theory Appl., 181 (2019), 706-707.  doi: 10.1007/s10957-018-01463-x.

[23]

B. Kohli, Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints, RAIRO Oper. Res., 53 (2019), 1617-1632.  doi: 10.1051/ro/2018084.

[24]

X. F. Li and J. Z. Zhang, Necessary optimality conditions in terms of convexificators in Lipschitz optimization, J. Optim. Theory Appl., 131 (2006), 429-452.  doi: 10.1007/s10957-006-9155-z.

[25]

D. V. Luu, Optimality condition for local efficient solutions of vector equilibrium problems via convexificators and applications, J. Optim. Theory Appl., 171 (2016), 643-665.  doi: 10.1007/s10957-015-0815-8.

[26]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I. Basic Theory, Fundamental Principles of Mathematical Sciences, 330, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31247-1.

[27]

B. S. Mordukhovich and N. M. Nam, Variational stability and marginal functions via generalized differentiation, Math. Oper. Res., 30 (2005), 800-816.  doi: 10.1287/moor.1050.0147.

[28]

B. S. MordukhovichN. M. Nam and N. D. Yen, Subgradients of marginal functions in parametric mathematical programming, Math. Program., 116 (2009), 369-396.  doi: 10.1007/s10107-007-0120-x.

[29]

J. V. Outrata, Necessary optimality conditions for Stackelberg problems, J. Optim. Theory Appl., 76 (1993), 305-320.  doi: 10.1007/BF00939610.

[30]

S. K. Suneja and B. Kohli, Optimality and duality results for bilevel programming problem using convexifactors, J. Optim. Theory Appl., 150 (2011), 1-19.  doi: 10.1007/s10957-011-9819-1.

[31]

S. K. Suneja and B. Kohli, Generalized nonsmooth cone convexity in terms of convexifactors in vector optimization, Opsearch, 50 (2013), 89-105.  doi: 10.1007/s12597-012-0092-3.

[32]

S. K. Suneja and B. Kohli, Duality for multiobjective fractional programming problem using convexifactors, Math. Sci. (Springer), 7: 6 (2013), 8pp. doi: 10.1186/2251-7456-7-6.

[33]

J. J. Ye, Nondifferentiable multiplier rules for optimization and bilevel optimization problems, SIAM J. Optim., 15 (2004), 252-274.  doi: 10.1137/S1052623403424193.

[34]

J. J. Ye, Constraint qualifications and KKT conditions for bilevel programming problems, Math. Oper. Res., 31 (2006), 811-824.  doi: 10.1287/moor.1060.0219.

[35]

J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optimization, 33 (1995), 9-27.  doi: 10.1080/02331939508844060.

show all references

References:
[1]

I. AhmadK. KummariV. Singh and A. Jayswal, Optimality and duality for nonsmooth minimax programming problems using convexifactors, Filomat, 31 (2017), 4555-4570.  doi: 10.2298/FIL1714555A.

[2]

J. F. Bard, Practical Bilevel Optimization. Algorithms and Applications, Nonconvex Optim. Appl., 30, Kluwer Acad. Publ., Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.

[3]

J. F. Bard, Optimality conditions for the bilevel programming problem, Naval Res. Logist. Quart., 31 (1984), 13-26.  doi: 10.1002/nav.3800310104.

[4]

J. F. Bard, Some properties of the bilevel programming problem, J. Optim. Theory Appl., 68 (1991), 371-378.  doi: 10.1007/BF00941574.

[5]

C. R. Bector, S. Chandra and J. Dutta, Principles of Optimization Theory, Narosa Publishing House, 2005.

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.

[7]

S. Dempe, Foundations of Bilevel Programming, Nonconvex Optim. Appl., 61, Kluwer Acad. Publ., Dordrecht, 2002. doi: 10.1007/b101970.

[8]

S. Dempe, A necessary and a sufficient optimality condition for bilevel programming problems, Optimization, 25 (1992), 341-354.  doi: 10.1080/02331939208843831.

[9]

S. Dempe, First-order necessary optimality conditions for general bilevel programming problems, J. Optim. Theory Appl., 95 (1997), 735-739.  doi: 10.1023/A:1022646611097.

[10]

S. DempeJ. Dutta and B. S. Mordukhovich, New necessary optimality conditions in optimistic bilevel programming, Optimization, 56 (2007), 577-604.  doi: 10.1080/02331930701617551.

[11]

V. F. Demyanov, Convexification and concavification of positively homogeneous function by the same family of linear functions, Report 3,208,802 from Universita di Pisa, 1994.

[12]

V. F. Demyanov and A. M. Rubinov, An introduction to quasidifferential calculus, in Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., 43, Kluwer Acad. Publ., Dordrecht, 2000, 1–31. doi: 10.1007/978-1-4757-3137-8_1.

[13]

J. Dutta and S. Chandra, Convexifactors, generalized convexity and optimality conditions, J. Optim. Theory Appl., 113 (2002), 41-64.  doi: 10.1023/A:1014853129484.

[14]

J. Dutta and S. Chandra, Convexifactors, generalized convexity and vector optimization, Optimization, 53 (2004), 77-94.  doi: 10.1080/02331930410001661505.

[15]

A. JayswalK. Kummari and V. Singh, Duality for a class of nonsmooth multiobjective programming problems using convexifactors, Filomat, 31 (2017), 489-498.  doi: 10.2298/FIL1702489J.

[16]

A. Jayswal, I. Stancu-Minasian and J. Banerjee, Optimality conditions and duality for interval-valued optimization problems using convexifactors, Rend. Circ. Mat. Palermo (2), 65 (2016), 17–32. doi: 10.1007/s12215-015-0215-9.

[17]

V. Jeyakumar and D. T. Luc, Nonsmooth calculus, maximality and monotonicity of convexificators, J. Optim. Theory Appl., 101 (1999), 599-621.  doi: 10.1023/A:1021790120780.

[18]

A. Kabgani and M. Soleimani-damaneh, Relationships between convexificators and Greensberg-Pierskalla subdifferentials for quasiconvex functions, Numer. Funct. Anal. Optim., 38 (2017), 1548-1563.  doi: 10.1080/01630563.2017.1349144.

[19]

A. KabganiM. 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: 10.1007/s00186-017-0584-2.

[20]

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: 10.1080/02331934.2017.1393675.

[21]

B. Kohli, Optimality conditions for optimistic bilevel programming problem using convexifactors, J. Optim. Theory Appl., 152 (2012), 632-651.  doi: 10.1007/s10957-011-9941-0.

[22]

B. Kohli, A note on the paper "Optimality conditions for optimistic bilevel programming problem using convexifactors", J. Optim. Theory Appl., 181 (2019), 706-707.  doi: 10.1007/s10957-018-01463-x.

[23]

B. Kohli, Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints, RAIRO Oper. Res., 53 (2019), 1617-1632.  doi: 10.1051/ro/2018084.

[24]

X. F. Li and J. Z. Zhang, Necessary optimality conditions in terms of convexificators in Lipschitz optimization, J. Optim. Theory Appl., 131 (2006), 429-452.  doi: 10.1007/s10957-006-9155-z.

[25]

D. V. Luu, Optimality condition for local efficient solutions of vector equilibrium problems via convexificators and applications, J. Optim. Theory Appl., 171 (2016), 643-665.  doi: 10.1007/s10957-015-0815-8.

[26]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I. Basic Theory, Fundamental Principles of Mathematical Sciences, 330, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31247-1.

[27]

B. S. Mordukhovich and N. M. Nam, Variational stability and marginal functions via generalized differentiation, Math. Oper. Res., 30 (2005), 800-816.  doi: 10.1287/moor.1050.0147.

[28]

B. S. MordukhovichN. M. Nam and N. D. Yen, Subgradients of marginal functions in parametric mathematical programming, Math. Program., 116 (2009), 369-396.  doi: 10.1007/s10107-007-0120-x.

[29]

J. V. Outrata, Necessary optimality conditions for Stackelberg problems, J. Optim. Theory Appl., 76 (1993), 305-320.  doi: 10.1007/BF00939610.

[30]

S. K. Suneja and B. Kohli, Optimality and duality results for bilevel programming problem using convexifactors, J. Optim. Theory Appl., 150 (2011), 1-19.  doi: 10.1007/s10957-011-9819-1.

[31]

S. K. Suneja and B. Kohli, Generalized nonsmooth cone convexity in terms of convexifactors in vector optimization, Opsearch, 50 (2013), 89-105.  doi: 10.1007/s12597-012-0092-3.

[32]

S. K. Suneja and B. Kohli, Duality for multiobjective fractional programming problem using convexifactors, Math. Sci. (Springer), 7: 6 (2013), 8pp. doi: 10.1186/2251-7456-7-6.

[33]

J. J. Ye, Nondifferentiable multiplier rules for optimization and bilevel optimization problems, SIAM J. Optim., 15 (2004), 252-274.  doi: 10.1137/S1052623403424193.

[34]

J. J. Ye, Constraint qualifications and KKT conditions for bilevel programming problems, Math. Oper. Res., 31 (2006), 811-824.  doi: 10.1287/moor.1060.0219.

[35]

J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optimization, 33 (1995), 9-27.  doi: 10.1080/02331939508844060.

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