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Optimality conditions of robust convex multiobjective optimization via \(\epsilon\)-constraint scalarization and image space analysis. (English) Zbl 1469.90098

In this paper, the robust efficiency conditions associated with a class of convex multiobjective optimization problems via \(\epsilon\)-constraint scalarization are formulated and proved. Moreover, the theoretical results are illustrated with some examples.

MSC:

90C17 Robustness in mathematical programming
90C29 Multi-objective and goal programming
90C25 Convex programming
Full Text: DOI

References:

[1] Ben-Tal, A.; Ghaoui, LE; Nemirovski, A., Robust optimization (2009), Princeton: Princeton University Press, Princeton · Zbl 1221.90001
[2] Ben-Tal, A.; Nemirovski, A., Robust convex optimization, Math Oper Res, 23, 769-805 (1998) · Zbl 0977.90052
[3] Jeyakumar, V.; Li, GY., Strong duality in robust convex programming: complete characterizations, SIAM J Optim, 20, 3384-3407 (2010) · Zbl 1228.90075
[4] Kuroiwa, D.; Lee, GM., On robust multiobjective optimization, Vietnam J Math, 40, 305-317 (2012) · Zbl 1302.90199
[5] Ben-Tal, A.; Nemirovski, A., Robust optimization-methodology and applications, Math Program Ser B, 92, 453-480 (2002) · Zbl 1007.90047
[6] Wang, L.; Fang, M., Robust optimization model for uncertain multiobjective linear programs, J Inequal Appl, 1, 22 (2018) · Zbl 1388.90109
[7] Deb, K.; Gupta, H., Introducing robustness in multi-objective optimization, Evol Comput, 14, 4, 463-494 (2006)
[8] Ansari, QH; Köbis, E.; Sharma, PK., Characterizations of set relations with respect to variable domination structures via oriented distance function, Optim, 67, 9, 1389-1407 (2018) · Zbl 1404.49010
[9] Khan, AA; Tammer, C.; Zălinescu, C., Set-Valued optimization: an introduction with applications (2015), Berlin: Springer, Berlin · Zbl 1308.49004
[10] Soyster, A., Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper Res, 21, 1154-1157 (1973) · Zbl 0266.90046
[11] Chen, JW; Köbis, E.; Yao, JC., Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints, J Optim Theory Appl, 181, 2, 411-436 (2019) · Zbl 1451.90139
[12] Lee, JH; Lee, GM., On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems, Ann Oper Res, 269, 1, 419-438 (2018) · Zbl 1446.90143
[13] Ide, J.; Schöbel, A., Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts, OR Spectrum, 38, 1, 235-271 (2016) · Zbl 1336.90056
[14] Wei, HZ; Chen, RC; Li, SJ., A unified characterization of multiobjective robustness via separation, J Optim Theory Appl, 179, 1, 86-102 (2018) · Zbl 1409.90179
[15] Ansari, QH; Köbis, E.; Sharma, PK., Characterizations of multiobjective robustness via oriented distance function and image space analysis, J Optim Theory Appl, 181, 3, 817-839 (2019) · Zbl 1416.49017
[16] Carathéodory, C.Calculus of variations and partial differential equations of the first order. New York: Chelsea; 1982. Translation of the volume Variationsrechnung und Partielle Differential Gleichungen Erster Ordnung. B.G. Teubner, Berlin, 1935. · Zbl 0011.35603
[17] Giannessi, F., Constrained optimization and image space analysis, 1 (2005), New York: Springer, New York · Zbl 1082.49001
[18] Giannessi, F., Theorems of the alternative and optimality conditions, J Optim Theory Appl, 42, 331-365 (1984) · Zbl 0504.49012
[19] Dien, PH; Mastroeni, G.; Pappalardo, M., Regularity conditions for constrained extremum problems via image space, J Optim Theory Appl, 80, 19-37 (1994) · Zbl 0797.90089
[20] Li, J.; Huang, NJ., Image space analysis for variational inequalites with cone constraints and applications to traffic equilibria, Sci China Math, 55, 851-868 (2012) · Zbl 1266.90182
[21] Zhu, SK., Constrained extremum problems, regularity conditions and image space analysis. Part I: the scalar finite dimensional case, J Optim Theory Appl, 177, 3, 770-787 (2018) · Zbl 1398.49034
[22] Chen, JW; Huang, L.; Li, SJ., Separations and optimality of constrained multiobjective optimization via improvement sets, J Optim Theory Appl, 178, 794-823 (2018) · Zbl 1447.90054
[23] Chen, JW, Li, SJ, Yao, JC.Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. J Indust Manag Optim. 2019. doi:.
[24] Haimes, Y.; Lasdon, L.; Wismer, D., On a bicriterion formulation of the problems of integrated system identification and system optimization, IEEE Trans Systems, Man, Cybernet, 1, 296-297 (1971) · Zbl 0224.93016
[25] Chankong, V.; Haimes, Y., Multiobjective decision making: theory and methodology (1983), New York: North-Holland, New York · Zbl 0622.90002
[26] Ehrgott, M., Multicriteria optimization (2005), New York: Springer, New York · Zbl 1132.90001
[27] Fliege, J.; Werner, R., Robust multiobjective optimization and applications in portfolio optimization, European J Oper Res, 234, 422-433 (2013) · Zbl 1304.91191
[28] Ehrgott, M.; Ide, J.; Schöbel, A., Minmax robustness for multi-objective optimization problems, European J Oper Res, 239, 1, 17-31 (2014) · Zbl 1339.90296
[29] Klamroth, K.; Köbis, E.; Schöbel, A., A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals, Optimization, 62, 5, 649-671 (2013) · Zbl 1273.90135
[30] Klamroth, K.; Köbis, E.; Schöbel, A., A unified approach to uncertain optimization, European J Oper Res, 260, 403-420 (2017) · Zbl 1403.90534
[31] Zălinescu, C., Convex analysis in general vector spaces (2002), Singapore: World Scientific, Singapore · Zbl 1023.46003
[32] Rockafellar, RT., Convex analysis (1970), Princeton: Princeton University Press, Princeton · Zbl 0193.18401
[33] Jeyakumar, V.; Lee, GM; Dinh, N., Characterizations of solution sets of convex vector minimization problems, European J Oper Res, 174, 1380-1395 (2006) · Zbl 1103.90090
[34] Jeyakumar, V., Asymptotic dual conditions characterizing optimality for convex programs, J Optim Theory Appl, 93, 153-165 (1997) · Zbl 0901.90158
[35] Hiriart-Urruty, JB; Lemarechal, C., Convex analysis and minimization algorithms. Vols I and II (1993), Berlin: Springer-Verlag, Berlin · Zbl 0795.49002
[36] BHiriart-Urruty, J.ε-Subdifferential, In: Convex analysis and optimization. J.P. Aubin and R. Vinter, Eds. London: Pitman; 43-49, 1982.
[37] Li, C.; Ng, KF; Pong, TK., Constraint qualifications for convex inequality systems with applications in constrained optimization, SIAM J Optim, 19, 163-187 (2008) · Zbl 1170.90009
[38] Jeyakumar, V.; Lee, GM; Dinh, N., New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM J Optim, 14, 534-547 (2003) · Zbl 1046.90059
[39] Li, XB; Wang, S., Characterizations of robust solution set of convex programs with uncertain data, Optim Lett, 12, 1387-1402 (2018) · Zbl 1396.90060
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