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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

Bhawna Kohli

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.

Keywords: Bilevel programming problem, convexifactors, value function, constraint qualifications, optimality conditions.
Mathematics Subject Classification: 90C26, 90C46.
Citation: Bhawna Kohli. Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3209-3221. doi: 10.3934/jimo.2020114
References:
[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: 10.2298/FIL1714555A.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

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

[14]

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

[15]

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[19]

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: 10.1007/s00186-017-0584-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

B. S. Mordukhovich, N. 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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

show all references

References:
[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: 10.2298/FIL1714555A.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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: 10.1007/s00186-017-0584-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

B. S. Mordukhovich, N. 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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

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