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doi: 10.3934/jimo.2021161
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A study on vector variational-like inequalities using convexificators and application to its bi-level form

Department of Mathematics, University of Central Florida, 4000 Central Blvd, P.O Box 161364, Orlando, Florida 32816-1364, USA

* Corresponding author: Gayatri Pany

Received  October 2020 Revised  July 2021 Early access September 2021

Fund Project: The first author is supported by grant from Mohapatra Family Foundation and the College of Graduate Studies of the University of Central Florida

This paper deals with the weak versions of the vector variational-like inequalities, namely Stampacchia and Minty type under invexity in the framework of convexificators. The connection between both the problems along with the link to vector optimization problem are analyzed. An application to nonconvex mathematical programming has also been presented. Further, the bi-level version of these problems is formulated and a procedure to obtain the solution involving the auxiliary principle technique is described in detail. We have shown that the iterative algorithm with the help of which we get the approximate solution converges strongly to the exact solution of the problem.

Citation: Gayatri Pany, Ram N. Mohapatra. A study on vector variational-like inequalities using convexificators and application to its bi-level form. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021161
References:
[1]

Q. H. Ansari and M. Rezaei, Generalized vector variational-like inequalities and vector optimization in asplund spaces, Optimization, 62 (2013), 721-734.  doi: 10.1080/02331934.2012.669758.

[2]

T. Antczak, Mean value in invexity analysis, Nonlinear Analysis: Theory, Methods & Applications, 60 (2005), 1473-1484.  doi: 10.1016/j.na.2004.11.005.

[3]

M. BianchiI. V. Konnov and R. Pini, Lexicographic variational inequalities with applications, Optimization, 56 (2007), 355-367.  doi: 10.1080/02331930600819704.

[4]

O. ChadliQ. H. Ansari and S. Al-Homidan, Existence of solutions and algorithms for bilevel vector equilibrium problems: An auxiliary principle technique, Journal of Optimization Theory and Applications, 172 (2017), 726-758.  doi: 10.1007/s10957-017-1062-y.

[5]

S.-L. Chen and N.-J. Huang, Vector variational inequalities and vector optimization problems on hadamard manifolds, Optimization Letters, 10 (2016), 753-767.  doi: 10.1007/s11590-015-0896-1.

[6]

B. D. Craven and S. M. N. Islam, Dynamic optimization models in finance: Some extensions to the framework, models, and computation, Journal of Industrial & Management Optimization, 10 (2014), 1129-1146.  doi: 10.3934/jimo.2014.10.1129.

[7]

V. F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, Journal of Global Optimization, 10 (1997), 305-326.  doi: 10.1023/A:1008246130864.

[8]

F. Giannessi, On minty variational principle, in New Trends in Mathematical Programming, vol. 13, Kluwer Acad. Publ., Boston, MA, (1998), 93–99. doi: 10.1007/978-1-4757-2878-1_8.

[9]

S.-M. Guu and J. Li, Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 2847-2855.  doi: 10.1016/j.na.2009.01.137.

[10]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.

[11]

M. A. Hejazi and S. Nobakhtian, Optimality conditions for multiobjective fractional programming, via convexificators, Journal of Industrial & Management Optimization, 16 (2020), 623-631.  doi: 10.3934/jimo.2018170.

[12]

V. Jeyakumar and D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, Journal of Optimization Theory and Applications, 101 (1999), 599-621.  doi: 10.1023/A:1021790120780.

[13]

S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, Journal of Optimization Theory and Applications, 18 (1976), 445-454.  doi: 10.1007/BF00932654.

[14]

B. Kohli, Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem, Journal of Industrial & Management Optimization. doi: 10.3934/jimo.2020114.

[15]

S. Komlósi, Generalized monotonicity and generalized convexity, Journal of Optimization Theory and Applications, 84 (1995), 361-376.  doi: 10.1007/BF02192119.

[16]

V. Laha and S. K. Mishra, On vector optimization problems and vector variational inequalities using convexificators, Optimization, 66 (2017), 1837-1850.  doi: 10.1080/02331934.2016.1250268.

[17]

C. S. Lalitha and M. Mehta, Vector variational inequalities with cone-pseudomonotone bifunctions, Optimization, 54 (2005), 327-338.  doi: 10.1080/02331930500100254.

[18]

F. Lara, Optimality conditions for nonconvex nonsmooth optimization via global derivatives, Journal of Optimization Theory and Applications, 185 (2020), 134-150.  doi: 10.1007/s10957-019-01613-9.

[19]

S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications, 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.

[20]

B. B. Upadhyay, P. Mishra, R. N. Mohapatra and S. K. Mishra, On the applications of nonsmooth vector optimization problems to solve generalized vector variational inequalities using convexificators, in World Congress on Global Optimization, Springer, 2019,660–671. doi: 10.1007/978-3-030-21803-4_66.

[21]

H. WeiC. Chen and B. Wu, Vector network equilibrium problems with uncertain demands and capacity constraints of arcs, Optimization Letters, 15 (2021), 1113-1131.  doi: 10.1007/s11590-020-01610-2.

[22]

X. Q. Yang and C. J. Goh, On vector variational inequalities: Application to vector equilibria, Journal of Optimization Theory and Applications, 95 (1997), 431-443.  doi: 10.1023/A:1022647607947.

show all references

References:
[1]

Q. H. Ansari and M. Rezaei, Generalized vector variational-like inequalities and vector optimization in asplund spaces, Optimization, 62 (2013), 721-734.  doi: 10.1080/02331934.2012.669758.

[2]

T. Antczak, Mean value in invexity analysis, Nonlinear Analysis: Theory, Methods & Applications, 60 (2005), 1473-1484.  doi: 10.1016/j.na.2004.11.005.

[3]

M. BianchiI. V. Konnov and R. Pini, Lexicographic variational inequalities with applications, Optimization, 56 (2007), 355-367.  doi: 10.1080/02331930600819704.

[4]

O. ChadliQ. H. Ansari and S. Al-Homidan, Existence of solutions and algorithms for bilevel vector equilibrium problems: An auxiliary principle technique, Journal of Optimization Theory and Applications, 172 (2017), 726-758.  doi: 10.1007/s10957-017-1062-y.

[5]

S.-L. Chen and N.-J. Huang, Vector variational inequalities and vector optimization problems on hadamard manifolds, Optimization Letters, 10 (2016), 753-767.  doi: 10.1007/s11590-015-0896-1.

[6]

B. D. Craven and S. M. N. Islam, Dynamic optimization models in finance: Some extensions to the framework, models, and computation, Journal of Industrial & Management Optimization, 10 (2014), 1129-1146.  doi: 10.3934/jimo.2014.10.1129.

[7]

V. F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, Journal of Global Optimization, 10 (1997), 305-326.  doi: 10.1023/A:1008246130864.

[8]

F. Giannessi, On minty variational principle, in New Trends in Mathematical Programming, vol. 13, Kluwer Acad. Publ., Boston, MA, (1998), 93–99. doi: 10.1007/978-1-4757-2878-1_8.

[9]

S.-M. Guu and J. Li, Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 2847-2855.  doi: 10.1016/j.na.2009.01.137.

[10]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.

[11]

M. A. Hejazi and S. Nobakhtian, Optimality conditions for multiobjective fractional programming, via convexificators, Journal of Industrial & Management Optimization, 16 (2020), 623-631.  doi: 10.3934/jimo.2018170.

[12]

V. Jeyakumar and D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, Journal of Optimization Theory and Applications, 101 (1999), 599-621.  doi: 10.1023/A:1021790120780.

[13]

S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, Journal of Optimization Theory and Applications, 18 (1976), 445-454.  doi: 10.1007/BF00932654.

[14]

B. Kohli, Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem, Journal of Industrial & Management Optimization. doi: 10.3934/jimo.2020114.

[15]

S. Komlósi, Generalized monotonicity and generalized convexity, Journal of Optimization Theory and Applications, 84 (1995), 361-376.  doi: 10.1007/BF02192119.

[16]

V. Laha and S. K. Mishra, On vector optimization problems and vector variational inequalities using convexificators, Optimization, 66 (2017), 1837-1850.  doi: 10.1080/02331934.2016.1250268.

[17]

C. S. Lalitha and M. Mehta, Vector variational inequalities with cone-pseudomonotone bifunctions, Optimization, 54 (2005), 327-338.  doi: 10.1080/02331930500100254.

[18]

F. Lara, Optimality conditions for nonconvex nonsmooth optimization via global derivatives, Journal of Optimization Theory and Applications, 185 (2020), 134-150.  doi: 10.1007/s10957-019-01613-9.

[19]

S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications, 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.

[20]

B. B. Upadhyay, P. Mishra, R. N. Mohapatra and S. K. Mishra, On the applications of nonsmooth vector optimization problems to solve generalized vector variational inequalities using convexificators, in World Congress on Global Optimization, Springer, 2019,660–671. doi: 10.1007/978-3-030-21803-4_66.

[21]

H. WeiC. Chen and B. Wu, Vector network equilibrium problems with uncertain demands and capacity constraints of arcs, Optimization Letters, 15 (2021), 1113-1131.  doi: 10.1007/s11590-020-01610-2.

[22]

X. Q. Yang and C. J. Goh, On vector variational inequalities: Application to vector equilibria, Journal of Optimization Theory and Applications, 95 (1997), 431-443.  doi: 10.1023/A:1022647607947.

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