Abstract
This paper aims to investigate optimality conditions for a weakly Pareto solution to a robust multi-objective optimization problem with locally Lipschitzian data. We do this by using a minimax programming approach, namely, by establishing the necessary optimality condition for a (local) optimal solution to a robust minimax optimization problem under a suitable constraint qualification, we then employ it to arrive in the desired target. In addition, some duality results for both robust minimax optimization problems and robust multi-objective optimization problems are also provided.
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References
Bram, J. (1966). The Lagrange multiplier theorem for max-min with several constraints. SIAM Journal on Applied Mathematics, 14, 665–667.
Bector, C. R., Chandra, S., & Kumar, V. (1994). Duality for a class of minmax and inexact programming problem. Journal of Mathematical Analysis and Applications, 186(3), 735–746.
Ben-Tal, A., & Nemirovski, A. (1997). Stable truss topology design via semidefinite programming. SIAM Journal on Optimization, 7(4), 991–1016.
Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805.
Ben-Tal, A., & Nemirovski, A. (2008). Selected topics in robust convex optimization. Mathematical Programming, 112(1), 125–158.
Ben-Tal, A., El Ghaoui, L., & Nemirovski, A. (2009). Robust optimization. New Jersy: Princeton University Press.
Beck, A., & Ben-Tal, A. (2009). Duality in robust optimization: Primal worst equals dual best. Operations Research Letters, 37(1), 1–6.
Ben-Tal, A., den Hertog, D., & Vial, J. (2015). Deriving robust counterparts of nonlinear uncertain inequalities. Mathematical Programming, 149(1–2), 265–299.
Chen, J., Kobis, E., & Yao, J. C. (2019). Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. Journal of Optimization Theory and Applications, 181, 411–436.
Chuong, T. D. (2016). Optimality and duality for robust multiobjective optimization problems. Nonlinear Analysis, 134, 127–143.
Chuong, T. D., & Kim, D. S. (2014). Optimality conditions and duality in nonsmooth multiobjective optimization problems. Annals of Operations Research, 217(1), 117–136.
Chuong, T. D., & Kim, D. S. (2016). A class of nonsmooth fractional multiobjective optimization problems. Annals of Operations Research, 244(2), 367–383.
Chuong, T. D., & Kim, D. S. (2017). Nondifferentiable minimax programming problems with applications. Annals of Operations Research, 251(1–2), 73–87.
Chuong, T. D., & Kim, D. S. (2018). Normal regularity for the feasible set of semi-infinite multiobjective optimization problems with applications. Annals of Operations Research, 267(1–2), 81–99.
Clarke, F. H. (1983). Optimization and nonsmooth analysis. New York: Wiley.
Ehrogtt, M. (2005). Multicriteria optimization. Berlin: Springer.
Ehrogtt, M., Ide, J., & Schobel, A. (2014). Minmax robustness for multi-objective optimization problems. European Journal of Operational Research, 239, 17–31.
El Ghaoui, L., & Lebret, H. (1997). Robust solution to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18(4), 1035–1064.
El Ghaoui, L., Oustry, F., & Lebret, H. (1998). Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9(1), 33–52.
Falk, J. E. (1976). Exact solutions to inexact linear programs. Operations Research, 24(4), 783–787.
Hong, Z., Bae, K. D., & Kim, D. S. (2019). Optimality conditions in convex optimization with locally Lipschitz constraints. Optimization Letters, 13, 1059–1068.
Jiao, L. G., & Lee, J. H. (2021). Finding efficient solutions in robust multiple objective optimization with SOS-convex polynomial data. Annals of Operations Research, 296, 803–820.
Kim, D. S., Son, P. T., & Tuyen, N. V. (2019). On the existence of Pareto solutions for polynomial vector optimization problems. Mathematical Programming, 177, 321–341.
Kouvelis, P., & Yu, G. (1997). Robust discrete optimization and its applications. London: Kluwer Academic Publishers.
Lai, H. C., & Huang, T. Y. (2012). Nondifferentiable minimax fractional programming in complex spaces with parametric duality. Journal of Global Optimization, 53(2), 243–254.
Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation, I: Basic theory. Berlin: Springer.
Mordukhovich, B. S. (2018). Variational Analysis and Applications, Springer Monographs in Mathematics.
Son, T. Q., & Kim, D. S. (2013). \(\varepsilon \)-Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints. Journal of Global Optimization, 57, 447–465.
Soyster, A. L. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21(5), 1154–1157.
Singh, C. (1982). Convex programming with set-inclusive constraints and its applications to generalized linear and fractional programming. Journal of Optimization Theory and Applications, 38(1), 33–42.
Tanimoto, S. (1980). Nondifferentiable mathematical programming and convex-concave functions. Journal of Optimization Theory and Applications, 31(3), 331–342.
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This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1A2C1008672).
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Hong, Z., Bae, K.D. & Kim, D.S. Minimax programming as a tool for studying robust multi-objective optimization problems. Ann Oper Res 319, 1589–1606 (2022). https://doi.org/10.1007/s10479-021-04179-w
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DOI: https://doi.org/10.1007/s10479-021-04179-w
Keywords
- Multi-objective optimization
- Minimax programming
- Generalized convexity
- KKT optimality conditions
- Duality