In this paper, we consider a vector optimization problem involving locally Lipschitz generalized approximately convex functions and provide several concepts of approximate efficient solutions. We formulate approximate vector variational inequalities of Minty and Stampacchia type under the framework of Clarke subdifferentials and use these inequalities as a tool to characterize an approximate efficient solution of the vector optimization problem.
Keywords: Clarke subdifferentials, vector optimization, approximate vector variational inequalities
@article{RO_2021__55_S1_S2999_0,
author = {Joshi, Bhuwan Chandra},
title = {On generalized approximate convex functions and variational inequalities},
journal = {RAIRO. Operations Research},
pages = {S2999--S3008},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
doi = {10.1051/ro/2020141},
mrnumber = {4223175},
zbl = {1469.49010},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2020141/}
}
TY - JOUR AU - Joshi, Bhuwan Chandra TI - On generalized approximate convex functions and variational inequalities JO - RAIRO. Operations Research PY - 2021 SP - S2999 EP - S3008 VL - 55 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2020141/ DO - 10.1051/ro/2020141 LA - en ID - RO_2021__55_S1_S2999_0 ER -
%0 Journal Article %A Joshi, Bhuwan Chandra %T On generalized approximate convex functions and variational inequalities %J RAIRO. Operations Research %D 2021 %P S2999-S3008 %V 55 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2020141/ %R 10.1051/ro/2020141 %G en %F RO_2021__55_S1_S2999_0
Joshi, Bhuwan Chandra. On generalized approximate convex functions and variational inequalities. RAIRO. Operations Research, Tome 55 (2021), pp. S2999-S3008. doi: 10.1051/ro/2020141
[1] , and , On the subdifferentiability of difference of two functions and local minimization. Set Valued Anal. 16 (2008) 413–427. | MR | Zbl | DOI
[2] and , Techniques of Variational Analysis. Springer-Verlag (2005). | MR | Zbl
[3] and , On the Dini-Hadamard subdifferential of the difference of two functions. J. Global Optim. 50 (2011) 485–502. | MR | Zbl | DOI
[4] , Asymptotic analysis for proximal-type methods in vector variational inequality problems. Oper. Res. Lett. 43 (2015) 226–230. | MR | Zbl | DOI
[5] , Invex function and constrained local minima. Bull. Aust. Math. Soc. 24 (1981) 357–366. | MR | Zbl | DOI
[6] and , Approximate convexity and submonotonicity. J. Math. Anal. App. 291 (2004) 292–301. | MR | Zbl | DOI
[7] and , Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003). | MR | Zbl
[8] and , Engineering and economic applications of complementarity problems. SIAM Rev. 39 (1997) 669–713. | MR | Zbl | DOI
[9] , and , Error bounds for affine variational inequalities with second-order cone constraints. Oper. Res. Lett. 45 (2017) 456–460. | MR | Zbl | DOI
[10] , and , Approximate convexity in vector optimization. Bull. Aust. Math. Soc. 74 (2006) 207–218. | MR | Zbl | DOI
[11] , , and , Vector critical points and efficiency in vector optimization with Lipschitz functions. Optim. Lett. 10 (2016) 47–62. | MR | Zbl | DOI
[12] , On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. App. 80 (1981) 545–550. | MR | Zbl | DOI
[13] , Second-order invex functions in nonlinear programming. Optimization 61 (2012) 489–503. | MR | Zbl | DOI
[14] , and , Solving policy design problems: alternating direction method of multipliers-based methods for structured inverse variational inequalities. Eur. J. Oper. Res. 280 (2020) 417–427. | MR | Zbl | DOI
[15] , and , -subdifferential and -monotonicity. Nonlinear Anal. 33 (1998) 71–90. | MR | Zbl | DOI
[16] , -solutions in vector minimization problems. J. Optim. Theory App. 43 (1984) 265–276. | MR | Zbl | DOI
[17] and , Invexity and Optimization. In: Vol. 88 of Nonconvex Optimization and Its Applications. Springer, Berlin (2008). | MR | Zbl | DOI
[18] and , On approximately star-shaped functions and approximate vector variational inequalities. J. Optim. Theory App. 156 (2012) 278–293. | MR | Zbl | DOI
[19] and , On minty variational principle for nonsmooth vector optimization problems with approximate convexity. Optim. Lett. 10 (2016) 577–589. | MR | Zbl | DOI
[20] and , Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity 17 (2013) 1071–1083. | MR | Zbl
[21] , Invex sets. Stud. Cerc. Mat. 46 (1994) 529–532. | MR | Zbl
[22] and , Approximately convex functions and approximately monotonic operators. Nonlinear Anal. 66 (2007) 547–564. | MR | Zbl
[23] and , Semismoothness and directional subconvexity of functions. Pac. J. Optim. 3 (2007) 323–344. | MR | Zbl
[24] , and , Approximate convex functions. J. Nonlinear Convex Anal. 1 (2000) 155–176. | MR | Zbl
[25] , and , Invex functions and generalized convexity in multiobjective programming. J. Optim. Theory App. 98 (1998) 651–661. | MR | Zbl
[26] , The directional subdifferential of the difference of two convex functions. J. Global Optim. 49 (2011) 505–519. | MR | Zbl
[27] , and , A new extragradient-type method for mixed variational inequalities. Oper. Res. Lett. 43 (2015) 567–572. | MR | Zbl
[28] , and , Vector critical points and generalized quasi-efficient solutions in nonsmooth multi-objective programming. J. Inequal. App. (2017) 1–12. | MR | Zbl
[29] , Epsilon efficiency. J. Optim. Theory App. 49 (1986) 319–337. | MR | Zbl
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