On some vector variational inequalities and optimization problems

Savin Treanţă; Savin Treanţă

doi:10.3934/math.2022795

AIMS Mathematics

2022, Volume 7, Issue 8: 14434-14443. doi: 10.3934/math.2022795

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On some vector variational inequalities and optimization problems

Savin Treanţă ^{1,2
,
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1.
Department of Applied Mathematics, University Politehnica of Bucharest, 060042 Bucharest, Romania
2.
Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania

Received: 10 March 2022 Revised: 18 May 2022 Accepted: 18 May 2022 Published: 06 June 2022
MSC : 49K20, 49J21

This paper establishes connections between the solutions of some new vector controlled variational inequalities and (proper) efficient solutions of the corresponding multiobjective controlled variational problem. More precisely, under the assumptions of invexity and Fréchet differentiability of the involved curvilinear integral functionals, and by using the notion of invex set with respect to some given functions, we derive the characterization results.
- (proper) efficient solution,
- invex set,
- Fréchet differentiability,
- curvilinear integral
Citation: Savin Treanţă. On some vector variational inequalities and optimization problems[J]. AIMS Mathematics, 2022, 7(8): 14434-14443. doi: 10.3934/math.2022795

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Abstract

This paper establishes connections between the solutions of some new vector controlled variational inequalities and (proper) efficient solutions of the corresponding multiobjective controlled variational problem. More precisely, under the assumptions of invexity and Fréchet differentiability of the involved curvilinear integral functionals, and by using the notion of invex set with respect to some given functions, we derive the characterization results.

References

[1]	I. Ahmad, K. Kummari, S. Al-Homidan, Sufficiency and Duality for Nonsmooth Interval-Valued Optimization Problems via Generalized Invex-Infine Functions, J. Oper. Res. Soc. China, 2022. https://doi.org/10.1007/s40305-021-00381-6
[2]	T. Antczak, $(p, r)$-Invexity in multiobjective programming, Eur. J. Oper. Res., 152 (2004), 72–87. https://doi.org/10.1016/S0377-2217(02)00696-3 doi: 10.1016/S0377-2217(02)00696-3
[3]	T. Antczak, Exact penalty functions method for mathematical programming problems involving invex functions, Eur. J. Oper. Res., 198 (2009), 29–36. https://doi.org/10.1016/j.ejor.2008.07.031 doi: 10.1016/j.ejor.2008.07.031
[4]	M. Arana-Jiménez, V. Blanco, E. Fernández, On the fuzzy maximal covering location problem, Eur. J. Oper. Res., 283 (2019), 692–705. https://doi.org/10.1016/j.ejor.2019.11.036 doi: 10.1016/j.ejor.2019.11.036
[5]	A. M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618–630. https://doi.org/10.1016/0022-247X(68)90201-1 doi: 10.1016/0022-247X(68)90201-1
[6]	B. A. Ghaznavi-ghosoni, E. Khorram, On approximating weakly/properly efficient solutions in multi-objective programming, Math. Comput. Model., 54 (2011), 3172–3181. https://doi.org/10.1016/j.mcm.2011.08.013 doi: 10.1016/j.mcm.2011.08.013
[7]	F. Giannessi, Theorems of the alternative quadratic programs and complementarity problems. In: R. Cottle, F. Giannessi, J. Lions, Eds., Variational inequalities and complementarity problems, Wiley, Chichester, 1980,151–186.
[8]	M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545–550. https://doi.org/10.1016/0022-247X(81)90123-2 doi: 10.1016/0022-247X(81)90123-2
[9]	A. Jayswal, S. Choudhury, R. U. Verma, Exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities, J. Appl. Math. Comput., 45 (2014), 87–97. https://doi.org/10.1007/s12190-013-0712-y doi: 10.1007/s12190-013-0712-y
[10]	S. Jha, P. Das, S. Bandhyopadhyay, S. Treanţă, Well-posedness for multi-time variational inequality problems via generalized monotonicity and for variational problems with multi-time variational inequality constraints, J. Comput. Appl. Math., 407 (2022), 114033. https://doi.org/10.1016/j.cam.2021.114033 doi: 10.1016/j.cam.2021.114033
[11]	K. R. Kazmi, Existence of solutions for vector optimization, Appl. Math. Lett., 9 (1996), 19–22. https://doi.org/10.1016/0893-9659(96)00088-2 doi: 10.1016/0893-9659(96)00088-2
[12]	M. H. Kim, Relations between vector continuous-time program and vector variational-type inequality, J. Appl. Math. Comput., 16 (2004), 279–287. https://doi.org/10.1007/BF02936168 doi: 10.1007/BF02936168
[13]	A. Klinger, Improper solutions of the vector maximum problem, Oper. Res., 15 (1967), 570–572. https://doi.org/10.1287/opre.15.3.570 doi: 10.1287/opre.15.3.570
[14]	S. K. Mishra, S. Y. Wang, K. K. Lai, Nondifferentiable multiobjective programming under generalized d-univexity, Eur. J. Oper. Res., 160 (2005), 218–226. https://doi.org/10.1016/S0377-2217(03)00439-9 doi: 10.1016/S0377-2217(03)00439-9
[15]	Şt. Mititelu, S. Treanţă, Efficiency conditions in vector control problems governed by multiple integrals, J. Appl. Math. Comput., 57 (2018), 647–665. https://doi.org/10.1007/s12190-017-1126-z doi: 10.1007/s12190-017-1126-z
[16]	G. Ruiz-Garzón, R. Osuna-Gómez, A, Rufián-Lizana, Relationships between vector variational-like inequality and optimization problems, Eur. J. Oper. Res., 157 (2004), 113–119. https://doi.org/10.1016/S0377-2217(03)00210-8 doi: 10.1016/S0377-2217(03)00210-8
[17]	G. Ruiz-Garzón, R. Osuna-Gómez, A, Rufián-Lizana, B. Hernández-Jiménez, Optimality in continuous-time multiobjective optimization and vector variational-like inequalities, TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer; Sociedad dexe Estadística e Investigación Operativa, 23 (2015), 198–219. https://doi.org/10.1007/s11750-014-0334-z doi: 10.1007/s11750-014-0334-z
[18]	S. Treanţă, A necessary and sufficient condition of optimality for a class of multidimensional control problems, Optim. Control Appl. Meth., 41 (2020), 2137–2148. https://doi.org/10.1002/oca.2645 doi: 10.1002/oca.2645
[19]	S. Treanţă, On Controlled Variational Inequalities Involving Convex Functionals, In: Le Thi H., Le H., Pham Dinh T., Eds, Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, Springer, Cham., 991 (2020), 164–174. https://doi.org/10.1007/978-3-030-21803-4_17
[20]	S. Treanţă, On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion, Eur. J. Control, 51 (2020), 1–9. https://doi.org/10.1016/j.ejcon.2019.07.003 doi: 10.1016/j.ejcon.2019.07.003
[21]	S. Treanţă, On well-posed isoperimetric-type constrained variational control problems, J. Differ. Equ., 298 (2021), 480–499. https://doi.org/10.1016/j.jde.2021.07.013 doi: 10.1016/j.jde.2021.07.013
[22]	S. Treanţă, Robust saddle-point criterion in second-order PDE & PDI constrained control problems, Int. J. Robust Nonlin. Control., 31 (2021), 9282–9293. https://doi.org/10.1002/rnc.5767 doi: 10.1002/rnc.5767

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