• Previous Article
    Coordinating a supply chain with demand information updating
  • JIMO Home
  • This Issue
  • Next Article
    The comparison between selling and leasing for new and remanufactured products with quality level in the electric vehicle industry
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  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 & Management Optimization, 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.  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. DempeJ. 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. 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.  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. 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.  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. 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.  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. 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.  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. DempeJ. 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. 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.  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. 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.  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. 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.  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

[1]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[2]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[3]

Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021013

[4]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391

[5]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[6]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[7]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[8]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

[9]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083

[10]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162

[11]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[12]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[13]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[14]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052

[15]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380

[16]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003

[17]

Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020050

[18]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[19]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[20]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (57)
  • HTML views (197)
  • Cited by (0)

Other articles
by authors

[Back to Top]