Abstract
In this paper, we establish robust optimality conditions for weakly efficient and properly positive efficient solutions of a semi-infinite multiobjective optimization with data uncertainty in both of the objective and constraints functions. Our conditions are form of the robust Karush–Kuhn–Tucker multipliers rules. To this aim, a basic and Pshenichnyi–Levin–Valadire constraint qualifications are proposed. Sufficient optimality conditions for these robust efficient solutions are also derived under the generalized convex assumptions. Furthermore, we also prove that the basic constraint qualification ensuring the boundedness of these robust multipliers sets. Some examples are provided to illustrate the presented results.
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References
Anitescu M (2000) Degenerate nonlinear programming with a quadratic growth condition. SIAM J Optim 10:1116–1135
Ben-Tal A, Nemirovski A (2008) Selected topics in robust convex optimization. Math Program (Ser B) 112:125–158
Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton
Bertsimas D, Brown DB (2009) Constructing uncertainty sets for robust linear optimization. Oper Res Lett 57:1483–1495
Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53:464–501
Birge JR, Wets R (1991) Stochastic programming I, II. J.C. Baltzer AG, Amsterdam
Bonnans JF, Shapiro A (2000) Perturbation analysis of optimization problems. Springer, New York
Chen J, Köbis E, Yao JC (2019) Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J Optim Theory Appl 181:411–436
Chuong TD (2016) Optimality and duality for robust multiobjective optimization problems. Nonlinear Anal 134:127–143
Chuong TD (2020) Robust optimality and duality in multiobjective optimization problems under data uncertainty. SIAM J Optim 30:1501–1526
Clarke FH (1983) Optimization and nonsmooth analysis. Wiley, New York
Durea M, Dutta J, Tammer C (2008) Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension. J Math Anal Appl 348:589–606
Dutta J, Lalitha CS (2006) Bounded sets of KKT multipliers in vector optimization. J Global Optim 36:425–437
Ehrgott M, Ide J, Schobel A (2014) Minmax robustness for multi-objective optimization problems. Eur J Oper Res 239:17–31
Gauvin J (1977) A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math Program 12:136–138
Goberna MA, Jeyakumar V, Li G, Lopez MA (2013) Robust linear semi-infinite programming duality under uncertainty. Math Program (Ser B) 139:185–203
Goberna MA, Jeyakumar V, Li G, Vicente-Perez J (2015) Robust solutions to multi-objective linear programs with uncertain data. Eur J Oper Res 242:730–743
Goberna MA, Jeyakumar V, Li G, Vicente-Perez J (2018) Guaranteeing highly robust weakly efficient solutions for uncertain multi-objective convex programs. Eur J Oper Res 270:40–50
Hiriart-Urruty JB, Lemarechal C (1991) Convex analysis and minimization algorithms. Springer, Berlin
Ide J, Schobel A (2016) Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts. OR Spectr 38:235–271
Jeyakumar V, Li G (2010) Strong duality in robust convex programming: complete characterizations. SIAM J Optim 20:3384–3407
Jeyakumar V, Li G, Lee GM (2012) Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Anal 75:1362–1373
Kuroiwa D, Lee GM (2012) On robust multiobjective optimization. Vietnam J Math 40:305–317
Kuroiwa D, Lee GM (2014) On robust convex multiobjective optimization. J. Nonlinear Convex Anal 15:1125–1136
Lee JH, Lee GM (2018) On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems. Ann Oper Res 269:419–438
Lee GM, Son PT (2014) On nonsmooth optimality theorems for robust optimization problems. Bull Korean Math Soc 51:287–301
Mordukhovich BS, Nam MN (2014) An easy path to convex analysis and applications. Morgan & Claypool, Williston, p 2014
Pourkarimi L, Soleimani-damaneh M (2016) Robustness in deterministic multi-objective linear programming with respect to the relative interior and angle deviation. Optimization 65:1983–2005
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Rockafellar RT, Wets JB (1998) Variational Analysis. Springer, Berlin
Tung NM, Duy MV (2002) Constraint qualifications and optimality conditions for robust nonsmooth semi-infinite multiobjective optimization problems. 4OR (to appear)
Wets JB (1989) Stochastic programming. Optimization. In: Nemhauser GL, Rinnooy AHG, Todd MJ (eds) Handbooks of operations research and management science, vol 1. North-Holland, Amsterdam, pp 573–629
Zamani M, Soleimani-damaneh M, Kabgani A (2015) Robustness in nonsmooth nonlinear multi-objective programming. Eur J Oper Res 247:370–378
Acknowledgements
The author wish to thank the Editors and the referee for their helpful comments and suggestions that helped us significantly improve the paper. This work was supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under grant no. 101.01-2021.13.
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Communicated by Gabriel Haeser.
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Tung, N.M. On robust Karush–Kuhn–Tucker multipliers rules for semi-infinite multiobjective optimization with data uncertainty. Comp. Appl. Math. 42, 98 (2023). https://doi.org/10.1007/s40314-023-02224-x
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DOI: https://doi.org/10.1007/s40314-023-02224-x