### Robust Optimization: Concise Review of Current Status and Issues

Park, Kyungchul

• 투고 : 2015.04.04
• 심사 : 2015.06.22
• 발행 : 2015.11.30
• 13 28

#### 초록

This paper gives a brief review of the theory and applications of the robust optimization. Major issues including tractability, relations with relevant optimization frameworks, and dynamic robust optimization are addressed and future research directions are presented.

#### 키워드

Robust Optimization;Stochastic Optimization;Survey

#### 참고문헌

1. Artzner, P., F. Delbaen, J. Eber, and D. Heath, "Coherent measures of risk," Mathematical Finance 9 (1999), 203-228. https://doi.org/10.1111/1467-9965.00068
2. Baron, O., J. Milner, and Naseraldin, "Facility location: A robust optimization approach," Production and Operations Management 20 (2011), 772-785. https://doi.org/10.1111/j.1937-5956.2010.01194.x
3. Ben-Tal, A., D. Bertsimas, and D. B. Brown, "A soft robust model for optimization under ambiguity," Operations Research 58 (2010), 1220-1234. https://doi.org/10.1287/opre.1100.0821
4. Ben-Tal, A., S. Bhadra, C., Bhattacharyya, and J. S. Nath, "Chance constrained uncertain classification via robust optimization," Mathematical Programming 127 (2011), 145-173. https://doi.org/10.1007/s10107-010-0415-1
5. Ben-Tal, A., A. Goryashko, E. Guslitzer, and A. Nemirovski, "Adjustable robust solutions of uncertain linear programs," Math. Programming 99 (2003), 351-376.
6. Ben-Tal, A. and A. Nemirovski, "Robust convex optimization," Math. Oper. Res. 23 (1998), 769-805. https://doi.org/10.1287/moor.23.4.769
7. Ben-Tal, A. and A. Nemirovski, "Robust solutions of uncertain linear programs," Operations Research Letters 25, 1 (1999), 1-13. https://doi.org/10.1016/S0167-6377(99)00016-4
8. Ben-Tal, A. and A. Nemirovski, "Robust solutions of linear programming problems contaminated with uncertain data," Math. Programming 88 (2000), 411-421. https://doi.org/10.1007/PL00011380
9. Ben-Tal, A. and A. Nemirovski, "Lectures on Modern Convex Optimization," SIAM 2001.
10. Ben-Tal, A. and A. Nemirovski, "On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty," SIAM Journal on Optimization 12 (2002), 811-833. https://doi.org/10.1137/S1052623400374756
11. Ben-Tal, A., A. Nemirovski, and C. Roos, "Robust solutions of uncertain quadratic and conic-quadratic problems," SIAM Journal on Optimization 13, 2 (2002), 535-560. https://doi.org/10.1137/S1052623401392354
12. Ben-Tal, A., B. Golany, A. Nemirovski, and J. P. Vial, "Supplier-retailer flexible commitments contracts: A robust optimization approach," Manufacturing and Service Operations Management 7, 3 (2005), 248-271. https://doi.org/10.1287/msom.1050.0081
13. Bertsimas, D. and D. B. Brown, "Constructing uncertainty sets for robust linear optimization," Operations Research 57, 6 (2009), 1483-1495. https://doi.org/10.1287/opre.1080.0646
14. Bertsimas, D., D. Brown, and, C. Caramanis, "Theory and applications of robust optimization," SIAM Review 53 (2011), 464-501. https://doi.org/10.1137/080734510
15. Bertsimas, D., V. Gupta, and N. Kallus, "Data-driven robust optimization," Submitted to Operations Research 2013.
16. Bertsimas, D., D. Pachamanova, and M. Sim, "Robust linear optimization under general norms," Operations Research Letters 32 (2004), 510-516. https://doi.org/10.1016/j.orl.2003.12.007
17. Bertsimas, D. and I. Popescu, "Optimal inequalities in probability theory: A convex optimization approach," SIAM Journal of Optimization 15, 3 (2004), 780-800.
18. Bertsimas, D. and M. Sim, "Robust discrete optimization and network flows," Mathematical Programming Series B 98 (2003), 49-71. https://doi.org/10.1007/s10107-003-0396-4
19. Bertsimas, D. and M. Sim, "The price of robustness," Operations Research 52, 1 (2004), 35-53. https://doi.org/10.1287/opre.1030.0065
20. Bertsimas, D. and M. Sim, "Tractable approximations to robust conic optimization problems," Mathematical Programming 107, 1 (2006), 5-36. https://doi.org/10.1007/s10107-005-0677-1
21. Bertsimas, D. and A. Thiele, "A robust optimization approach to supply chain management," Operations Research 54, 1 (2006), 150-168. https://doi.org/10.1287/opre.1050.0238
22. Birge, J. R. and F. Louveaux, "Introduction to Stochastic Programming," Springer-Verlag 1997.
23. Boyd, S., L. El Ghaoui, E. Feron, and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory," Society for Industrial and Applied Mathematics 1994.
24. Boyd, S. and L. Vandenberghe, "Semidefinite programming," SIAM Review 38, 1 (1996), 49-95. https://doi.org/10.1137/1038003
25. Chen, X., M. Sim, and P. Sun, "A robust optimization perspective to stochastic programming," Operations Research 55, 6 (2007), 1058-1071. https://doi.org/10.1287/opre.1070.0441
26. Erera, A. L., J. C. Morales, and M. Savelsbergh, "Robust optimization for empty repositioning problems," Operations Research 57, 2 (2009), 468-483. https://doi.org/10.1287/opre.1080.0650
27. Fabozzi, F. J., D. Huang, and G. Zhou, "Robust portfolios: contributions from operations research and finance," Annals of Operations Research 176 (2010), 191-220. https://doi.org/10.1007/s10479-009-0515-6
28. Gabrel, V., C. Murat, and A Thiele, "Recent advances in robust optimization: An overview," European Journal of Operational Research 235 (2014), 471-483. https://doi.org/10.1016/j.ejor.2013.09.036
29. Gao, C., E. Johnson, and B. Smith, "Integrated airline fleet and crew robust planning," Transportation Science 43 (2009), 2-16. https://doi.org/10.1287/trsc.1080.0257
30. Ghaoui, L. and H. Lebret, "Robust solutions to least-squares problems with uncertain data," SIAM J. Matrix Analysis and Applications 18, 4 (1997), 1035-1064. https://doi.org/10.1137/S0895479896298130
31. Ghaoui, E., M. Oks, and F. Oustry, "Worst-case value-at-risk and robust portfolio optimization: A conic programming approach," Operations Research 51, 4 (2003), 543-556. https://doi.org/10.1287/opre.51.4.543.16101
32. Ghaoui, L., F. Oustry, and H. Lebret. "Robust solutions to uncertain semidefinite programs," SIAM J. Optimization 9, 1 (1998), 33-52. https://doi.org/10.1137/S1052623496305717
33. Goldfarb, D. and G. Iyengar, "Robust portfolio selection problems," Mathematics of Operations Research 1 (2003), 1-38.
34. Jiang, R., J. Wang, and Y. Guan, "Robust unit commitment with wind power and pumped storage hydro," IEEE Transactions on Power Systems 27 (2012), 800-810. https://doi.org/10.1109/TPWRS.2011.2169817
35. Kouvelis, P. and G. Yu, "Robust Discrete Optimization and Its Applications," Kluwer Academic Publishers Norwell, MA, 1997.
36. Lee, C., Lee, K. Park, and S. Park, "Technical note-Branch-and-price-and-cut approach to the robust network design problem without flow bifurcations," Operations Research 60, 3 (2012), 604-610. https://doi.org/10.1287/opre.1120.1049
37. Lorenz, R. and S. Boyd. "Robust minimum variance beam forming," IEEE Transactions on Signal Processing 53, 5 (2005), 1684-1696. https://doi.org/10.1109/TSP.2005.845436
38. Natarajan, K., D. Pachamanova, and M. Sim, "Constructing risk measures from uncertainty sets," Operations Research 57, 5(2009), 1129-1141. https://doi.org/10.1287/opre.1080.0683
39. Nemirovski, A., "Several NP-hard problems arising in robust stability analysis," Math. Control Signals Systems 6 (1993), 99-105. https://doi.org/10.1007/BF01211741
40. Popescu, I., "Robust mean-covariance solutions for stochastic optimization," Operations Research 55, 1 (2007), 98-112. https://doi.org/10.1287/opre.1060.0353
41. Rusmevichientong, P. and H. Topaloglu, "Robust assortment optimization in revenue management under the multinomial logit choice model," Operations Research 60 (2012), 865-882. https://doi.org/10.1287/opre.1120.1063
42. Xu, H., C. Caramanis, and S. Mannor, "A distributional interpretation of robust optimization," In Proceedings of the Allerton Conference on Communication, Control and Computing, 2010.
43. Xu, H., C. Caramanis, and S. Mannor, "Robust regression and Lasso," IEEE Transactions on Information Theory 56, 7 (2010), 3561-3574. https://doi.org/10.1109/TIT.2010.2048503