On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems
Tóm tắt
In this paper, we consider a semi-infinite multiobjective optimization problem with more than two differentiable objective functions and uncertain constraint functions, which is called a robust semi-infinite multiobjective optimization problem and give its robust counterpart
$${\mathrm{(RSIMP)}}$$
of the problem, which is regarded as the worst case of the uncertain semi-infinite multiobjective optimization problem. We prove a necessary optimality theorem for a weakly robust efficient solution of
$${\mathrm{(RSIMP)}} $$
, and then give a sufficient optimality theorem for a weakly robust efficient solution of
$${\mathrm{(RSIMP)}}$$
. We formulate a Wolfe type dual problem of
$${\mathrm{(RSIMP)}}$$
and give duality results which hold between
$${\mathrm{(RSIMP)}}$$
and its dual problem.
Tài liệu tham khảo
Aubin, J.-P. (1979). Mathematical Methods of Game and Economic Theory. Amsterdam: North-Holland Publishing Company.
Beck, A., & Ben-Tal, A. (2009). Duality in robust optimization: Primal worst equals dual best. Operations Research Letters, 37, 1–6.
Ben-Tal, A., Ghaoui, L. E., & Nemirovski, A. (2009). Robust optimzation. Princeton series in applied mathematics. Priceton, NJ: Priceton University Press.
Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions to uncertain linear programs. Operations Research Letters, 25, 1–13.
Ben-Tal, A., & Nemirovski, A. (2002). Robust optimization-methodology and applications. Mathematical programming, series B, 92, 453–480.
Ben-Tal, A., & Nemirovski, A. (2008). A selected topics in robust convex optimization. Mathematical programming, series B, 112, 125–158.
Bonnans, J. F., & Shapiro, A. (2000). Perturbation Analysis of Optimization Problems. New York: Springer.
Chuong, T. D. (2016). Optimality and duality for robust multiobjective optimization Problems. Nonlinear Analysis, 134, 127–143.
Chuong, T. D., & Kim, D. S. (2014). Nonsmooth semi-infinite multiobjective optimization problems. Journal of Optimization Theory and Applications, 160, 748–762.
Goberna, M. A., Guerra-Vazquez, F., & Todorov, M. I. (2013). Constraint qualifications in linear vector semi-infinite optimization. European Journal of Operational Research, 227, 12–21.
Goberna, M. A., Jeyakumar, V., Li, G., & Lopez, M. A. (2013). Robust linear semi-infinite programming duality under uncertainty. Mathematical programming, series B, 139, 185–203.
Goberna, M. A., Jeyakumar, V., Li, G., & Vicente-Perez, J. (2014). Robust solutions of multi-objective linear semi-infinite programs under constraint data uncertainty. SIAM Journal on Optimization, 24, 1402–1419.
Goberna, M. A., & Lopez, M. A. (1998). Linear semi-infinite optimization. Chichester: Wiley.
Jeyakumar, V., Lee, G. M., & Li, G. (2015). Characterizing robust solutions sets convex programs under data uncertainty. Journal of Optimization Theory and Applications, 64, 407–435.
Jeyakumar, V., & Li, G. (2010). Characterizing robust set containments and solutions of uncertain linear programs without qualifications. Operations Research Letters, 38, 188–194.
Jeyakumar, V., & Li, G. (2010). Strong duality in robust convex programming: Complete characterizations. SIAM Journal on Optimization, 20, 3384–3407.
Jeyakumar, V., & Li, G. (2014). Robust semi-definite linear programming duality under data uncertainty. Optimization, 63, 713–733.
Jeyakumar, V., Li, G., & Lee, G. M. (2012). Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Analysis, 75, 1362–1373.
Kuroiwa, D., & Lee, G. M. (2012). On robust multiobjective optimization. Vietnam Journal of Mathematics, 40, 305–317.
Kuroiwa, D., & Lee, G. M. (2014). On robust convex multiobjective optimization. Journal of Nonlinear and Convex Analysis, 15, 1125–1136.
Lee, J. H., & Jiao, L. (2016). On quasi \(\epsilon \)-solution for robust convex optimization problems. Optimization Letters. doi:10.1007/s11590-016-1067-8.
Lee, J. H., & Lee, G. M. (2012). On \(\epsilon \)-solutions for convex optimization problems with uncertainty data. Positivity, 16, 509–526.
Lee, G. M., & Lee, J. H. (2015). On nonsmooth optimality theorems for robust multiobjective optimization problems. Journal of Nonlinear and Convex Analysis, 16, 2039–2052.
Lee, J. H., & Lee, G. M. (2015). On sequential optimality conditions for robust multiobjective convex optimization problems. Linear and Nonlinear Analysis, 1, 221–246.
Li, G. Y., Jeyakumar, V., & Lee, G. M. (2011). Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Analysis, 74, 2327–2341.
Suzuki, S., Kuroiwa, D., & Lee, G. M. (2013). Surrogate duality for robust optimization. European Journal of Operational Research, 231, 257–262.