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Selected topics in robust convex optimization

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Abstract

Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-but- bounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control.

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References

  1. Adida E. and Perakis G. (2006). A robust optimization approach to dynamic pricing and inventory control with no backorders. Math. Program. 107: 97–129

    Article  MATH  MathSciNet  Google Scholar 

  2. Atamturk, A., Zhang, M.: Two-stage robust network flow and design under demand uncertainty. Research Report BCOL.04.03, IEOR, University of California-Berkeley. December 2004

  3. Atamturk, A.: Strong formulations of robust mixed 0-1 programming. Research Report BCOL.03.04, IEOR, University of California-Berkeley. December 2003. Revised February 2005

  4. Beck A., Ben-Tal A. and Eldar Y. (2006). Robust mean-squared error estimation of multiple signals in linear systems affected by model and noise uncertainties. Math. Program. 107: 155–187

    Article  MATH  MathSciNet  Google Scholar 

  5. Ben-Tal A. and Nemirovski A. (1997). Stable truss topology design via semidefinite programming. SIAM J. Optim. 7: 991–1016

    Article  MATH  MathSciNet  Google Scholar 

  6. Ben-Tal A. and Nemirovski A. (1998). Robust convex optimization. Math. Oper. Res. 23: 769–805

    MATH  MathSciNet  Google Scholar 

  7. Ben-Tal A. and Nemirovski A. (1999). Robust solutions to uncertain linear programs. OR Lett. 25: 1–13

    MATH  MathSciNet  Google Scholar 

  8. Ben-Tal A. and Nemirovski A. (2000). Structural design via semidefinite programming. In: Saigal, R., Wolkowitcz, H. and Vandenberghe, L. (eds) Handbook on Semidefinite Programming., pp. Kluwer, Dordrecht

    Google Scholar 

  9. Ben-Tal A. and Nemirovski A. (2000). Robust solutions of Linear Programming problems contaminated with uncertain data. Math. Program. 88: 411–424

    Article  MATH  MathSciNet  Google Scholar 

  10. Ben-Tal, A., El-Ghaoui, L., Nemirovski, A.: Robust semidefinite programming. In: Saigal, R., Vandenberghe, L., Wolkowicz, H (eds.) Semidefinite Programming and Applications. Kluwer Dordrecht (2000)

  11. Ben-Tal, A., Margalit, T., Nemirovski, A.: Robust Modeling of Multi-Stage Portfolio Problems. In: Frenk, H., Roos, K., Terlaky, T., Zhang, S.(eds.) High Performance Optimization, Dordrecht Kluwer pp. 303–328 (2000)

  12. Ben-Tal A. and Nemirovski A. (2001). Lectures on Modern Convex Optimization. SIAM, Philadelphia

    MATH  Google Scholar 

  13. Ben-Tal A. and Nemirovski A. (2002). Robust optimization—methodology and applications. Math. Program. Ser. B 92: 453–480

    Article  MATH  MathSciNet  Google Scholar 

  14. Ben-Tal A. and Nemirovski A. (2002). On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM J. Optim. 12: 811–833

    Article  MATH  MathSciNet  Google Scholar 

  15. Ben-Tal A., Nemirovski A. and Roos C. (2002). Robust solutions of uncertain quadratic and conic-quadratic problems. SIAM J. Optim. 13: 535–560

    Article  MATH  MathSciNet  Google Scholar 

  16. Ben-Tal A., Nemirovski A. and Roos C. (2003). Extended Matrix Cube Theorems with applications to μ-Theory in Control. Math. Oper. Res. 28: 497–523

    Article  MATH  MathSciNet  Google Scholar 

  17. Ben-Tal A., Goryashko A., Guslitzer E. and Nemirovski A. (2004). Adjustable robust solutions of uncertain linear programs. Math. Program. 99: 351–376

    Article  MATH  MathSciNet  Google Scholar 

  18. Ben-Tal A., Golany B., Nemirovski A. and Vial J.-Ph. (2005). Supplier-retailer flexible commitments contracts: a robust optimization approach. Manuf. Serv. Oper. Manage. 7(3): 248–273

    Article  Google Scholar 

  19. Ben-Tal, A., Boyd, S., Nemirovski, A.: Extending the scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Program. Ser. B, Special issue on Robust Optimization. (2006) (to appear) E-print: http://www.optimization-online.org/DB_HTML/2005/05/1136.html

  20. Ben-Tal, A., Boyd, S., Nemirovski, A.: Control of uncertainty-affected discrete time linear systems via convex programming. SIAM J. Control Optim. (submitted); E-print: http://www.optimization-online.org/DB_HTML/2005/10/1232.html

  21. Bertsimas, D., Sim, M.: Robust discrete optimization under ellipsoidal uncertainty sets. Technical Report, MIT, April 2004

  22. Bertsimas D. and Sim M. (2003). Robust discrete optimization and network flows. Math. Program. Ser. B, 98: 49–71

    Article  MATH  MathSciNet  Google Scholar 

  23. Bertsimas D. and Sim M. (2004). The price of robustness. Oper. Res. 52(1): 35–53

    Article  MathSciNet  Google Scholar 

  24. Bertsimas D., Pachamanova D. and Sim M. (2004). Robust linear optimization under general norms. Oper. R. Lett. 32(6): 510–516

    Article  MATH  MathSciNet  Google Scholar 

  25. Bertsimas D. and Thiele A. (2006). Robust optimization approach to inventory theory. Oper. Res. 54: 150–168

    Article  MathSciNet  Google Scholar 

  26. Bertsimas D. and Sim M. (2006). Tractable approximations to robust conic optimization problems. Math. Program. 107: 5–36

    Article  MATH  MathSciNet  Google Scholar 

  27. Bhattachrrya S., Grate L., Mian S., El Ghaoui L. and Jordan M. (2004). Robust sparse hyperplane classifiers: application to uncertain molecular profiling data. J. Comput. Bio. 11(6): 1073–1089

    Article  Google Scholar 

  28. Boyd S., El Ghaoui L., Feron E. and Balakrishnan V. (1994). Linear matrix inequalities in system and control theory. SIAM, Philadelphia

    MATH  Google Scholar 

  29. Calafiore, G., El Ghaoui, L.: Worst-case maximum likelihood estimation in the linear model. Automatica 37(4) (2001)

  30. Charnes A., Cooper W.W. and Symonds G.H. (1958). Cost horizons and certainty equivalents: an to stochastic programming of heating oil. Manage. Sci. 4: 235–263

    Google Scholar 

  31. Dan Barb F., Ben-Tal A. and Nemirovski A. (2003). Robust dissipativity of interval uncertain system. SIAM J. Control Optim. 41: 1661–1695

    Article  MATH  MathSciNet  Google Scholar 

  32. Diehl M., Bock H.-G. and Kostina E. (2006). An approximation technique for robust nonlinear optimization. Math. Program. 107: 213–230

    Article  MATH  MathSciNet  Google Scholar 

  33. Eldar Y., Ben-Tal A. and Nemirovski A. (2005). Robust mean-squared error estimation in the presence of model uncertainties. IEEE Trans. Signal Process. 53: 168–181

    Article  MathSciNet  Google Scholar 

  34. El-Ghaoui L. and Lebret H. (1997). Robust solutions to least-square problems with uncertain data matrices. SIAM J. Matrix Anal. Appl. 18: 1035–1064

    Article  MATH  MathSciNet  Google Scholar 

  35. El-Ghaoui L., Oustry F. and Lebret H. (1998). Robust solutions to uncertain semidefinite programs.. SIAM J. Optim. 9: 33–52

    Article  MATH  MathSciNet  Google Scholar 

  36. El Ghaoui L., Oks M. and Oustry F. (2003). Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51(4): 543–556

    Article  MathSciNet  Google Scholar 

  37. El Ghaoui, L.: Inversion error, condition number, and approximate inverses of uncertain matrices. Linear Algebra Appl. 343–344, 171–193 (2002)

  38. El Ghaoui L. and Calafiore G. (2001). Robust filtering for discrete-time systems with bounded noise and parametric uncertainty. IEEE Trans. Automat. Control 46: 1084–1089

    Article  MATH  MathSciNet  Google Scholar 

  39. Goldfarb D. and Iyengar G. (2003). Robust portfolio selection problems. Math. Oper. Res. 28(1): 1–37

    Article  MATH  MathSciNet  Google Scholar 

  40. Goldfarb D. and Iyengar G. (2003). Robust quadratically constrained programs. Math. Program. Ser. B 97(3): 495–515

    Article  MATH  MathSciNet  Google Scholar 

  41. Goulart, P.J., Kerrigan, E.C., Maciejowski, J.M.: Optimization over state feedback policies for robust control with constraints. Technical Report CUED/F-INFENG/TR.494, Cambridge University Engineering Department, March 2005. E-print: http://www-control.eng.cam.ac.uk/, Automatica (to appear)

  42. Guslitser, E.: Uncertatinty-immunized solutions in linear programming. Master Thesis, Technion, Israeli Institute of Technology, IE&M faculty (2002). E-print: http://iew3.technion.ac.il/Labs/Opt/index.php?4

  43. Iyengar G. (2005). Robust dynamic programming. Math. Oper. Res. 30(2): 1–21

    Article  MathSciNet  Google Scholar 

  44. Iyengar G. and Erdogan E. (2006). Ambiguous chance constrained problems and robust optimization. Math. Program. 107: 17–31

    MathSciNet  Google Scholar 

  45. Kočvara M., Zowe J. and Nemirovski A. (2000). Cascading—an approach to robust material optimization. Comput. Struct. 76: 431–442

    Article  Google Scholar 

  46. Kostyukova O. and Kostina E. (2006). Robust optimal feedback for terminal linear-quadratic control problems under disturbances. Math. Program. 107: 131–153

    Article  MATH  MathSciNet  Google Scholar 

  47. Lanckriet G.R.G., El Ghaoui L., Bhattacharyya C. and Jordan M. (2002). A robust minimax approach to classification. J. Mach. Learn. Res. 3: 555–582

    Article  MathSciNet  Google Scholar 

  48. Lasserre J.B. (2006). Robust global optimization with polynomials. Math. Program. 107: 275–293

    Article  MATH  MathSciNet  Google Scholar 

  49. Miller L.B. and Wagner H. (1965). Chance-constrained programming with joint constraints. Oper. Res. 13: 930–945

    MATH  Google Scholar 

  50. Nemirovski, A.: Regular Banach spaces and large deviations of random sums. Paper in progress, E-print: http://www2.isye.gatech.edu/~nemirovs/

  51. Nemirovski A. and Shapiro A. (2006). Convex approximations of chance constrained programs. SIAM J. Optim. 17: 969–996

    Article  MATH  MathSciNet  Google Scholar 

  52. Genin Y., Hachez Y., Nesterov Yu. and Van Dooren P. (2003). Optimization problems over positive pseudopolynomial matrices.. SIAM J. Matrix Anal. Appl. 25: 57–79

    Article  MATH  MathSciNet  Google Scholar 

  53. Nilim, A., El Ghaoui, L., Duong, Vu.: Algorithms for multi-aircraft re-routing under uncertainties. In: Actes de la Deuxieme Conference Internationale Associant Chercheurs Vietnamiens et Francophones en Informatique, Hanoï Vietnam, 2-5 Février 2004 (RIVF 2004)

  54. Nilim A. and El Ghaoui L. (2005). Robust control of Markov decision processes with uncertain transition matrices. Oper. Res. 53: 780–798

    Article  MathSciNet  Google Scholar 

  55. Perakis G. and Sood A. (2006). Competitive multi-period pricing for perishable products: a robust optimization approach. Math. Program. 107: 295–335

    Article  MATH  MathSciNet  Google Scholar 

  56. Pinter J. (1989). Deterministic approximations of probability inequalities.. ZOR - Methods Models Oper. Res. Ser. Theory 33: 219–239

    Article  MATH  MathSciNet  Google Scholar 

  57. Prékopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton Symposium on Mathematical Programming, pp. 113–138. Princeton University Press, Princeton (1970)

  58. Prékopa A. (1995). Stochastic Programming. Kluwer, Dordrecht

    Google Scholar 

  59. Prékopa, A., Vizvari, B., Badics, T.: Programming under probabilistic constraint with discrete random variables. In: Grandinetti, L. et al. (eds.) New Trends in Mathematical Programming, pp 235–257 Kluwer, Dordrecht (1997)

  60. Prékopa, A.: Probabilistic programming. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming. Handbook in OR & MS, Vol. 10, North-Holland Publishing Company, Amsterdam (2003)

  61. Scherer C.W. and Hol C.W.J. (2006). Matrix sum-of-squares relaxations for robust semi-definite programs. Math. Program. 107: 189–211

    Article  MATH  MathSciNet  Google Scholar 

  62. Soyster A.L. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21: 1154–1157

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, Technion – Israel Institute of Technology, Technion city, Haifa, 32000, Israel

    Aharon Ben-Tal

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0205, USA

    Arkadi Nemirovski

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  1. Aharon Ben-Tal
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  2. Arkadi Nemirovski
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Correspondence to Arkadi Nemirovski.

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Ben-Tal, A., Nemirovski, A. Selected topics in robust convex optimization. Math. Program. 112, 125–158 (2008). https://doi.org/10.1007/s10107-006-0092-2

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