# OPTIMALITY CONDITIONS AND DUALITY IN NONDIFFERENTIABLE ROBUST OPTIMIZATION PROBLEMS

• Kim, Moon Hee (Department of Refrigeration Engineering, Tongmyong University)
• Accepted : 2015.02.23
• Published : 2015.05.31

#### Abstract

We consider a nondifferentiable robust optimization problem, which has a maximum function of continuously differentiable functions and support functions as its objective function, continuously differentiable functions as its constraint functions. We prove optimality conditions for the nondifferentiable robust optimization problem. We formulate a Wolfe type dual problem for the nondifferentiable robust optimization problem and prove duality theorems.

#### References

1. D. Bertsimas, D. Brown, Constructing uncertainty sets for robust linear optimization, Oper. Res. 57(2009), 1483-1495. https://doi.org/10.1287/opre.1080.0646
2. A. Ben-Tal, A. Nemirovski, Robust-optimization-methodology and applications, Math. Program., Ser B 92(2002), 453-480. https://doi.org/10.1007/s101070100286
3. A. Ben-Tal, A. Nemirovski, A selected topics in robust convex optimization, Math. Program., Ser B 112(2008), 125-158.
4. D. Bertsimas, D. Pachamanova, M. Sim, Robust linear optimization under general norms, Oper. Res. Lett. 32(2004), 510-516. https://doi.org/10.1016/j.orl.2003.12.007
5. A. Ben-Tal, L. E. Ghaoui, A. Nemirovski, Robust optimization, Princeton Series in Applied Mathematics, 2009.
6. V. Jeyakumar, G. Li, G. M. Lee, A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty, Oper. Res. Lett. 39(2011), 109-114. https://doi.org/10.1016/j.orl.2011.02.007
7. V. Jeyakumar, G. Li, G. M. Lee, Robust duality for generalized convex programming problems under data uncertainty, Nonlinear Analysis 75(2012), 1362-1373. https://doi.org/10.1016/j.na.2011.04.006
8. M. H. Kim, Robust duality for generalized invex programming problems, Commun. Korean Math. Soc. 28(2013), 419-423. https://doi.org/10.4134/CKMS.2013.28.2.419
9. D. Kuroiwa and G. M. Lee, On robust multiobjective optimization, Vietnam J. Math. 40(2012), 305-317.
10. G. M. Lee and M. H. Kim, On duality theorems for robust optimization problems, Journal of the Chungcheong Mathematical Society 26(2013), 723-734. https://doi.org/10.14403/jcms.2013.26.4.723
11. A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2009.