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Zhou, Zhiang; Huang, Min; Köbis, Elisabeth

Globally proper efficiency of set optimization problems based on the certainly set less order relation. (English) Zbl 1539.90085

Appl. Anal. 103, No. 1, 184-197 (2024).

MSC:

90C26 Nonconvex programming, global optimization
90C29 Multi-objective and goal programming
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming

Keywords:

set optimization; globally proper efficient solution; Lagrangian multiplier; duality; saddle point
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References:

[1] Adán, M, Novo, V.Proper efficiency in vector optimization on real linear spaces. J Optim Theory Appl. 2004;121:515-540. · Zbl 1076.90051
[2] Ansari, QH, Köbis, E, Yao, J-C.Vector variational inequalities and vector optimization. Berlin: Springer; 2018. (). · Zbl 1394.49001
[3] Chen, GY, Rong, WD.Characterizations of the Benson proper efficiency for nonconvex vector optimization. J Optim Theory Appl. 1998;98:365-384. · Zbl 0908.90225
[4] Gutiérrez, C, Huerga, L, Jiménez, B, et al. Approximate solutions of vector optimization problems via improvement sets in real linear spaces. J Glob Optim. 2018;70:875-901. · Zbl 1441.90145
[5] Li, ZF.Benson proper efficiency in the vector optimization of set-valued maps. J Optim Theory Appl. 1998;98:623-649. · Zbl 0913.90235
[6] Zhou, ZA, Yang, XM, Peng, JW.ε-Henig proper efficiency of set-valued optimization problems in real ordered linear spaces. Optim Lett. 2014;8:1813-1827. · Zbl 1297.90132
[7] Xia, LY, Qiu, JH.Superefficiency in vector optimization with nearly subconvexlike set-valued maps. J Optim Theory Appl. 2008;136:125-137. · Zbl 1194.90093
[8] Qiu, QS, Yang, XM.Connectedness of Henig weakly efficient solution set for set-valued optimization problems. J Optim Theory Appl. 2012;152:439-449. · Zbl 1250.90079
[9] Khan, AA, Tammer, C, Zălinescu, C.Set-valued optimization – an introduction with applications. Berlin, Heidelberg: Springer; 2015. · Zbl 1308.49004
[10] Ansari, QH, Hussain, N, Sharma, PK.Convergence of the solution sets for set optimization problems. J Nonlinear Var Anal. 2022;6:165-183. · Zbl 07556350
[11] Jahn, J.Vectorization in set optimization. J Optim Theory Appl. 2015;167:783-795. · Zbl 1368.90142
[12] Jahn, J.Vectorization in nonconvex set optimization. J Appl Numer Optim. 2022;4:19-36.
[13] Köbis, E, Köbis, MA.Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization. Optimization. 2016;65:1805-1827. · Zbl 1384.90092
[14] Gerlach, T, Rocktäschel, S.On convexity and quasiconvexity of extremal value functions in set optimization. Appl Set-Valued Anal Optim. 2021;3:293-308.
[15] Kuroiwa, D, Tanaka, T, Ha, TXD.On cone convexity of set-valued maps. Nonlinear Anal. 1997;30:1487-1496. · Zbl 0895.26010
[16] Hernández, E, Rodríguez-Marín, L.Lagrangian duality in set-valued optimization. J Optim Theory Appl. 2007;134:119-134. · Zbl 1129.49029
[17] Dhingra, M, Lalitha, CS.Set optimization using improvement sets. Yugoslav J Oper Res. 2017;27:153-167. · Zbl 1474.49036
[18] Chicco, M, Miggnanego, F, Pusillo, L, et al. Vector optimization problems via improvement sets. J Optim Theory Appl. 2011;150:516-529. · Zbl 1231.90329
[19] Hernández, E, Rodríguez-Marín, L.Existence theorems for set optimization problems. Nonlinear Anal. 2007;67:1726-1736. · Zbl 1278.90379
[20] Kuroiwa, D.Existence theorems of set optimization with set-valued maps. J Inf Optim Sci. 2003;24:73-84. · Zbl 1176.90545
[21] Han, Y.Nonlinear scalarizing functions in set optimization problems. Optimization. 2019;68:1685-1718. · Zbl 1431.90142
[22] Hernández, E, Rodríguez-Marín, L.Nonconvex scalarization in set optimization with set-valued maps. J Math Anal Appl. 2007;325:1-18. · Zbl 1110.90080
[23] Han, Y.Connectedness of weak minimal solution set for set optimization problems. Oper Res Lett. 2020;48:820-826. · Zbl 1525.49027
[24] Han, Y, Wang, SH, Huang, NJ.Arcwise connectedness of the solution sets for set optimization problems. Oper Res Lett. 2019;47:168-172. · Zbl 1476.90349
[25] Ansari, QH, Sharma, PK. Set order relations, set optimization, and ekelands variational principle. In: Laha V, Marchal P, Mishra SK, editors. Optimization, variational analysis and applications. Singapore: Springer; 2021. p. 103-165. (IFSOVAA 2020. Springer proceedings in mathematics and statistics; vol. 355).
[26] Jahn, J, Ha, TXD.New order relations in set optimization. J Optim Theory Appl. 2011;148:209-236. · Zbl 1226.90092
[27] Yang, XM, Yang, XQ, Chen, GY.Theorems of the alternative and optimization with set-valued maps. J Optim Theory Appl. 2000;107:627-640. · Zbl 0966.90071
[28] Pan, MM, Qiu, QS.The global proper efficiency for set-valued optimization problems via improvement sets. J Zhejiang Norm Univ (Nat Sci). 2018;41:375-382. · Zbl 1438.90273
[29] Luc, DT.Theory of vector optimization. In: Beekmann, M, Krelle, W, editors. Lecture notes in economics and mathematical systems, Vol. 319. Berlin: Springer; 1989.
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