Abstract
This paper devotes to the quasi \(\epsilon \)-solution (one sort of approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish approximate optimality theorem and approximate duality theorems in term of Wolfe type on quasi \(\epsilon \)-solution for the robust convex optimization problem. Moreover, some examples are given to illustrate the obtained results.
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Ben-Tal, A., Nemirovski, A.: A selected topics in robust convex optimization. Math. Progr. Ser. B. 112, 125–158 (2008)
Boyd, S., Vandemberghe, L.: Convex optimization. Cambridge Univ. Press, Cambridge (2004)
Chuong, T.D., Kim, D.S.: Approximate solutions of multiobjective optimization problems. Positivity 20, 187–207 (2016)
Dutta, J.: Necessary optimality conditions and saddle points for approximate optimization in Banach spaces. Top 13, 127–143 (2005)
Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. J. Glob. Optim. 56, 1463–1499 (2013)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Houda, M.: Comparison of approximations in stochastic and robust optimization programs. In: Hušková, M., Janžura, M. (eds.) Prague stochastics 2006, pp. 418–425. Matfyzpress, Prague (2006)
Jeyakumar, V.: Asymptotic dual conditions characterizing optimality for convex programs. J. Optim. Theory Appl. 93, 153–165 (1997)
Jeyakumar, V., Lee, G.M., Dinh, N.: Characterization of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174, 1380–1395 (2006)
Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14, 534–547 (2003)
Jeyakumar, V., Li, G.Y.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)
Lee, J.H., Lee, G.M.: On \(\epsilon \)-solutions for convex optimization problems with uncertainty data. Positivity 16, 509–526 (2012)
Loridan, P.: Necessary conditions for \(\epsilon \)-optimality. Math. Progr. Stud. 19, 140–152 (1982)
Rockafellar, R.T.: Convex analysis. Princeton Univ. Press, Princeton (1970)
Rockafellar, R.T., Wets, R.J.B.: Variational analysis. Springer, Berlin (2001)
Son, T.Q., Strodiot, J.J., Nguyen, V.H.: \(\epsilon \)-optimality and \(\epsilon \)-Lagrangian duality for a nonconvex programming problem with an infinite number of constraints. J. Optim. Theory Appl. 141, 389–409 (2009)
Sun, X.K., Peng, Z.Y., Guo, X.L.: Some characterizations of robust optimal solutions for uncertain convex optimization problems. Optim. Lett. (2015). doi:10.1007/s11590-015-0946-8
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The authors would like to express their sincere thanks to anonymous referees for variable suggestions and comments for the paper.
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Lee, J.H., Jiao, L. On quasi \(\epsilon \)-solution for robust convex optimization problems. Optim Lett 11, 1609–1622 (2017). https://doi.org/10.1007/s11590-016-1067-8
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DOI: https://doi.org/10.1007/s11590-016-1067-8
Keywords
- Robust convex optimization problem
- Quasi \(\epsilon \)-solution
- Approximate optimality conditions
- Approximate duality theorems