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EURO Journal on Computational Optimization

, Volume 6, Issue 3, pp 211–238 | Cite as

Robust combinatorial optimization under convex and discrete cost uncertainty

  • Christoph Buchheim
  • Jannis KurtzEmail author
Original Paper

Abstract

In this survey, we discuss the state of the art of robust combinatorial optimization under uncertain cost functions. We summarize complexity results presented in the literature for various underlying problems, with the aim of pointing out the connections between the different results and approaches, and with a special emphasis on the role of the chosen uncertainty sets. Moreover, we give an overview over exact solution methods for NP-hard cases. While mostly concentrating on the classical concept of strict robustness, we also cover more recent two-stage optimization paradigms.

Keywords

Robust optimization Uncertainty Combinatorial optimization Two-stage robustness K-Adaptability Complexity 

Mathematics Subject Classification

90C99 

References

  1. Adjiashvili D, Zenklusen R (2011) An s–t connection problem with adaptability. Discrete Appl Math 159(8):695–705CrossRefGoogle Scholar
  2. Adjiashvili D, Stiller S, Zenklusen R (2015) Bulk-robust combinatorial optimization. Math Program 149(1–2):361–390CrossRefGoogle Scholar
  3. Adjiashvili D, Bindewald V, Michaels D (2016) Robust assignments via ear decompositions and randomized rounding. In: Chatzigiannakis I, Mitzenmacher M, Rabani Y, Sangiorgi D (eds) 43rd international colloquium on automata, languages, and programming (ICALP 2016). Leibniz international proceedings in informatics (LIPIcs), vol 55. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, pp. 71:1–71:14.  https://doi.org/10.4230/LIPIcs.ICALP.2016.71
  4. Adjiashvili D, Bindewald V, Michaels D (2017) Robust assignments with vulnerable nodes. Technical report. http://arxiv.org/abs/1703.06074
  5. Aissi H, Bazgan C, Vanderpooten D (2005a) Approximation complexity of min–max (regret) versions of shortest path, spanning tree, and knapsack. In: Algorithms—ESA 2005. Lecture notes in computer science, vol 3669. Springer, Berlin, pp 862–873Google Scholar
  6. Aissi H, Bazgan C, Vanderpooten D (2005b) Complexity of the min–max and min–max regret assignment problems. Oper Res Lett 33(6):634–640CrossRefGoogle Scholar
  7. Aissi H, Bazgan C, Vanderpooten D (2005c) Complexity of the min–max (regret) versions of cut problems. In: Algorithms and computation. Springer, Berlin, pp 789–798Google Scholar
  8. Aissi H, Bazgan C, Vanderpooten D (2005d) Pseudo-polynomial algorithms for min–max and min–max regret problems. In: 5th international symposium on operations research and its applications (ISORA 2005), pp 171–178Google Scholar
  9. Aissi H, Bazgan C, Vanderpooten D (2009) Min–max and min–max regret versions of combinatorial optimization problems: a survey. Eur J Oper Res 197(2):427–438CrossRefGoogle Scholar
  10. Álvarez-Miranda E, Ljubić I, Toth P (2013) A note on the Bertsimas & Sim algorithm for robust combinatorial optimization problems. 4OR 11(4):349–360.  https://doi.org/10.1007/s10288-013-0231-6 CrossRefGoogle Scholar
  11. Armon A, Zwick U (2006) Multicriteria global minimum cuts. Algorithmica 46(1):15–26CrossRefGoogle Scholar
  12. Atamtürk A (2006) Strong formulations of robust mixed 0–1 programming. Math Program 108:235–250CrossRefGoogle Scholar
  13. Atamtürk A, Narayanan V (2008) Polymatroids and mean-risk minimization in discrete optimization. Oper Res Lett 36(5):618–622CrossRefGoogle Scholar
  14. Atamtürk A, Zhang M (2007) Two-stage robust network flow and design under demand uncertainty. Oper Res 55(4):662–673CrossRefGoogle Scholar
  15. Averbakh I, Lebedev V (2004) Interval data minmax regret network optimization problems. Discrete Appl Math 138(3):289–301CrossRefGoogle Scholar
  16. Averbakh I, Lebedev V (2005) On the complexity of minmax regret linear programming. Eur J Oper Res 160(1):227–231.  https://doi.org/10.1016/j.ejor.2003.07.007 CrossRefGoogle Scholar
  17. Ayoub J, Poss M (2016) Decomposition for adjustable robust linear optimization subject to uncertainty polytope. Comput Manag Sci 13(2):219–239CrossRefGoogle Scholar
  18. Baumann F, Buchheim C, Ilyina A (2014) Lagrangean decomposition for mean–variance combinatorial optimization. In: Combinatorial optimization—third international symposium, ISCO 2014. lecture notes in computer science, vol 8596. Springer, Berlin, pp 62–74Google Scholar
  19. Baumann F, Buchheim C, Ilyina A (2015) A Lagrangean decomposition approach for robust combinatorial optimization. Technical report, Optimization onlineGoogle Scholar
  20. Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805CrossRefGoogle Scholar
  21. Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13CrossRefGoogle Scholar
  22. Ben-Tal A, Nemirovski A (2002) Robust optimization-methodology and applications. Math Program 92(3):453–480CrossRefGoogle Scholar
  23. Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2):351–376CrossRefGoogle Scholar
  24. Ben-Tal A, Golany B, Nemirovski A, Vial JP (2005) Retailer–supplier flexible commitments contracts: a robust optimization approach. Manuf Serv Oper Manag 7(3):248–271CrossRefGoogle Scholar
  25. Bertsimas D, Caramanis C (2010) Finite adaptability in multistage linear optimization. IEEE Trans Autom Control 55(12):2751–2766CrossRefGoogle Scholar
  26. Bertsimas D, Dunning I (2016) Multistage robust mixed-integer optimization with adaptive partitions. Oper Res 64(4):980–998CrossRefGoogle Scholar
  27. Bertsimas D, Georghiou A (2014) Binary decision rules for multistage adaptive mixed-integer optimization. Math Program 167:1–39Google Scholar
  28. Bertsimas D, Georghiou A (2015) Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Oper Res 63(3):610–627CrossRefGoogle Scholar
  29. Bertsimas D, Goyal V (2013) On the approximability of adjustable robust convex optimization under uncertainty. Math Methods Oper Res 77(3):323–343CrossRefGoogle Scholar
  30. Bertsimas D, Lubin IDM (2016) Reformulation versus cutting-planes for robust optimization. Comput Manag Sci 13(2):195–217CrossRefGoogle Scholar
  31. Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98(1–3):49–71CrossRefGoogle Scholar
  32. Bertsimas D, Sim M (2004a) The price of robustness. Oper Res 52(1):35–53CrossRefGoogle Scholar
  33. Bertsimas D, Sim M (2004b) Robust discrete optimization under ellipsoidal uncertainty sets. CiteseerGoogle Scholar
  34. Bertsimas D, Pachamanova D, Sim M (2004) Robust linear optimization under general norms. Oper Res Lett 32(6):510–516CrossRefGoogle Scholar
  35. Bertsimas D, Iancu DA, Parrilo PA (2010) Optimality of affine policies in multistage robust optimization. Math Oper Res 35(2):363–394CrossRefGoogle Scholar
  36. Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501CrossRefGoogle Scholar
  37. Beyer HG, Sendhoff B (2007) Robust optimization—a comprehensive survey. Comput Methods Appl Mech Eng 196(33):3190–3218CrossRefGoogle Scholar
  38. Billionnet A, Costa MC, Poirion PL (2014) 2-Stage robust MILP with continuous recourse variables. Discrete Appl Math 170:21–32CrossRefGoogle Scholar
  39. Buchheim C, Kurtz J (2016) Min–max–min robust combinatorial optimization subject to discrete uncertainty. Optimization onlineGoogle Scholar
  40. Buchheim C, Kurtz J (2017) Min–max–min robust combinatorial optimization. Math Program 163(1):1–23CrossRefGoogle Scholar
  41. Buchheim C, De Santis M, Rinaldi F, Trieu L (2015) A Frank–Wolfe based branch-and-bound algorithm for mean-risk optimization. Technical report, Optimization onlineGoogle Scholar
  42. Büsing C (2011) Recoverable robustness in combinatorial optimization. Ph.D. thesis, Technical University of BerlinGoogle Scholar
  43. Büsing C (2012) Recoverable robust shortest path problems. Networks 59(1):181–189CrossRefGoogle Scholar
  44. Büsing C, D’Andreagiovanni F (2012) New results about multi-band uncertainty in robust optimization. In: International symposium on experimental algorithms. Springer, Berlin, pp 63–74Google Scholar
  45. Büsing C, D’Andreagiovanni F (2013) Robust optimization under multi-band uncertainty—part I: theory. arXiv preprint arXiv:1301.2734
  46. Büsing C, Koster A, Kutschka M (2011a) Recoverable robust knapsacks: \(\gamma \)-scenarios. Network optimization. Springer, Berlin, pp 583–588Google Scholar
  47. Büsing C, Koster AM, Kutschka M (2011b) Recoverable robust knapsacks: the discrete scenario case. Optim Lett 5(3):379–392CrossRefGoogle Scholar
  48. Calafiore GC (2008) Multi-period portfolio optimization with linear control policies. Automatica 44(10):2463–2473CrossRefGoogle Scholar
  49. Chang TJ, Meade N, Beasley J, Sharaiha Y (2000) Heuristics for cardinality constrained portfolio optimisation. Comput Oper Res 27:1271–1302CrossRefGoogle Scholar
  50. Chassein A, Goerigk M (2016) Min–max regret problems with ellipsoidal uncertainty sets. arXiv preprint arXiv:1606.01180
  51. Chassein A, Goerigk M, Kasperski A, Zieliński P (2017) On recoverable and two-stage robust selection problems with budgeted uncertainty. arXiv preprint arXiv:1701.06064
  52. Chen X, Zhang Y (2009) Uncertain linear programs: extended affinely adjustable robust counterparts. Oper Res 57(6):1469–1482CrossRefGoogle Scholar
  53. Claßen G, Koster AM, Schmeink A (2015) The multi-band robust knapsack problem—a dynamic programming approach. Discrete Optim 18:123–149CrossRefGoogle Scholar
  54. Cornuejols G, Tütüncü R (2006) Optimization methods in finance, vol 5. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  55. Corporation I (2015) IBM ILOG CPLEX optimization studio: CPLEX user’s manual. https://www.ibm.com/us-en/marketplace/ibm-ilog-cplex
  56. El Ghaoui L, Lebret H (1997) Robust solutions to least-squares problems with uncertain data. SIAM J Matrix Anal Appl 18(4):1035–1064CrossRefGoogle Scholar
  57. El Ghaoui L, Oustry F, Lebret H (1998) Robust solutions to uncertain semidefinite programs. SIAM J Optim 9(1):33–52CrossRefGoogle Scholar
  58. Feige U, Jain K, Mahdian M, Mirrokni V (2007) Robust combinatorial optimization with exponential scenarios. In: Integer programming and combinatorial optimization, pp 439–453Google Scholar
  59. Fischetti M, Monaci M (2009) Light robustness. In: Ahuja RK, Möhring RH, Zaroliagis CD (eds) Robust and online large-scale optimization. Springer, Berlin, pp 61–84CrossRefGoogle Scholar
  60. Fischetti M, Monaci M (2012) Cutting plane versus compact formulations for uncertain (integer) linear programs. Math Program Comput 4:1–35CrossRefGoogle Scholar
  61. Fischetti M, Salvagnin D, Zanette A (2009) Fast approaches to improve the robustness of a railway timetable. Transp Sci 43(3):321–335CrossRefGoogle Scholar
  62. Gabrel V, Murat C, Thiele A (2014) Recent advances in robust optimization: an overview. Eur J Oper Res 235(3):471–483CrossRefGoogle Scholar
  63. Georghiou A, Wiesemann W, Kuhn D (2015) Generalized decision rule approximations for stochastic programming via liftings. Math Program 152(1–2):301–338CrossRefGoogle Scholar
  64. Gorissen BL, Yanıkoğlu İ, den Hertog D (2015) A practical guide to robust optimization. Omega 53:124–137CrossRefGoogle Scholar
  65. Grötschel M, Lovász L, Schrijver A (1993) Geometric algorithms and combinatorial optimization. Springer, BerlinCrossRefGoogle Scholar
  66. Gurobi Optimization I (2016) Gurobi optimizer reference manual. http://www.gurobi.com. Accessed 3 Sept 2018
  67. Hanasusanto GA, Kuhn D, Wiesemann W (2015) K-Adaptability in two-stage robust binary programming. Oper Res 63(4):877–891CrossRefGoogle Scholar
  68. Hradovich M, Kasperski A, Zieliński P (2016) The robust recoverable spanning tree problem with interval costs is polynomially solvable. arXiv preprint arXiv:1602.07422
  69. Iancu DA (2010) Adaptive robust optimization with applications in inventory and revenue management. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  70. Iancu DA, Sharma M, Sviridenko M (2013) Supermodularity and affine policies in dynamic robust optimization. Oper Res 61(4):941–956CrossRefGoogle Scholar
  71. Ilyina A (2017) Combinatorial optimization under ellipsoidal uncertainty. Ph.D. thesis, TU Dortmund UniversityGoogle Scholar
  72. Inuiguchi M, Sakawa M (1995) Minimax regret solution to linear programming problems with an interval objective function. Eur J Oper Res 86(3):526–536.  https://doi.org/10.1016/0377-2217(94)00092-Q CrossRefGoogle Scholar
  73. Kasperski A, Zieliński P (2011) On the approximability of robust spanning tree problems. Theor Comput Sci 412(4–5):365–374CrossRefGoogle Scholar
  74. Kasperski A, Zieliński P (2015) Robust recoverable and two-stage selection problems. arXiv preprint arXiv:1505.06893
  75. Kasperski A, Zieliński P (2016) Robust discrete optimization under discrete and interval uncertainty: a survey. In: Doumpos M, Zopounidis C, Grigoroudis E (eds) Robustness analysis in decision aiding, optimization, and analytics. Springer, Berlin, pp 113–143Google Scholar
  76. Kasperski A, Zieliński P (2017) Robust two-stage network problems. In: Operations research proceedings 2015. Springer, Berlin, pp 35–40Google Scholar
  77. Kasperski A, Kurpisz A, Zieliński P (2014) Recoverable robust combinatorial optimization problems. In: Operations research proceedings 2012. Springer, Berlin, pp 147–153Google Scholar
  78. Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, BerlinCrossRefGoogle Scholar
  79. Khandekar R, Kortsarz G, Mirrokni V, Salavatipour M (2008) Two-stage robust network design with exponential scenarios. Algorithms ESA 2008:589–600Google Scholar
  80. Kouvelis P, Yu G (1996) Robust discrete optimization and its applications. Springer, BerlinGoogle Scholar
  81. Kuhn D, Wiesemann W, Georghiou A (2011) Primal and dual linear decision rules in stochastic and robust optimization. Math Program 130(1):177–209CrossRefGoogle Scholar
  82. Kurtz J (2016) Min–max–min robust combinatorial optimization. Ph.D. thesis, TU Dortmund UniversityGoogle Scholar
  83. Lappas NH, Gounaris CE (2018) Robust optimization for decision-making under endogenous uncertainty. Comput Chem Eng 111:252–266CrossRefGoogle Scholar
  84. Lee T (2014) A short note on the robust combinatorial optimization problems with cardinality constrained uncertainty. 4OR 12(4):373–378.  https://doi.org/10.1007/s10288-014-0270-7 CrossRefGoogle Scholar
  85. Li Z, Ding R, Floudas CA (2011) A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization. Ind Eng Chem Res 50(18):10567–10603CrossRefGoogle Scholar
  86. Liebchen C, Lübbecke M, Möhring R, Stiller S (2009) The concept of recoverable robustness, linear programming recovery, and railway applications. In: Ahuja R, Möhring R, Zaroliagis C (eds) Robust and online large-scale optimization. Springer, Berlin, pp 1–27Google Scholar
  87. Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91Google Scholar
  88. Minoux M (2011) On 2-stage robust LP with RHS uncertainty: complexity results and applications. J Glob Optim 49(3):521–537CrossRefGoogle Scholar
  89. Mokarami S, Hashemi SM (2015) Constrained shortest path with uncertain transit times. J Glob Optim 63(1):149–163.  https://doi.org/10.1007/s10898-015-0280-9 CrossRefGoogle Scholar
  90. Monaci M, Pferschy U, Serafini P (2013) Exact solution of the robust knapsack problem. Comput Oper Res 40(11):2625–2631.  https://doi.org/10.1016/j.cor.2013.05.005 CrossRefGoogle Scholar
  91. Mutapcic A, Boyd S (2009) Cutting-set methods for robust convex optimization with pessimizing oracles. Optim Methods Softw 24(3):381–406CrossRefGoogle Scholar
  92. Naoum-Sawaya J, Buchheim C (2016) Robust critical node selection by benders decomposition. INFORMS J Comput 28(1):162–174.  https://doi.org/10.1287/ijoc.2015.0671 CrossRefGoogle Scholar
  93. Nasrabadi E, Orlin JB (2013) Robust optimization with incremental recourse. arXiv preprint arXiv:1312.4075
  94. Nikolova E (2010a) Approximation algorithms for offline risk-averse combinatorial optimization. Technical reportGoogle Scholar
  95. Nikolova E (2010b) Approximation algorithms for reliable stochastic combinatorial optimization. In: Serna M, Shaltiel R, Jansen K, Rolim J (eds) Approximation, randomization, and combinatorial optimization. Algorithms and techniques. Springer, Berlin, pp 338–351CrossRefGoogle Scholar
  96. Nohadani O, Sharma K (2016) Optimization under decision-dependent uncertainty. arXiv preprint arXiv:1611.07992
  97. Park KC, Lee KS (2007) A note on robust combinatorial optimization problem. Manag Sci Financ Eng 13(1):115–119Google Scholar
  98. Pessoa AA, Poss M (2015) Robust network design with uncertain outsourcing cost. INFORMS J Comput 27(3):507–524CrossRefGoogle Scholar
  99. Poss M (2013) Robust combinatorial optimization with variable budgeted uncertainty. 4OR 11(1):75–92CrossRefGoogle Scholar
  100. Poss M (2017) Robust combinatorial optimization with knapsack uncertainty. Discrete Optim 27:88–102CrossRefGoogle Scholar
  101. Postek K, den Hertog D (2016) Multistage adjustable robust mixed-integer optimization via iterative splitting of the uncertainty set. INFORMS J Comput 28(3):553–574CrossRefGoogle Scholar
  102. Saito H, Murota K (2007) Benders decomposition approach to robust mixed integer programming. Pac J Optim 3:99–112Google Scholar
  103. Schöbel A (2014) Generalized light robustness and the trade-off between robustness and nominal quality. Math Methods Oper Res 80:1–31CrossRefGoogle Scholar
  104. Shapiro A (2011) A dynamic programming approach to adjustable robust optimization. Oper Res Lett 39(2):83–87CrossRefGoogle Scholar
  105. Sim M (2004) Robust optimization. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  106. Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper Res 21(5):1154–1157CrossRefGoogle Scholar
  107. Subramanyam A, Gounaris CE, Wiesemann W (2017) K-Adaptability in two-stage mixed-integer robust optimization. arXiv preprint arXiv:1706.07097
  108. Vayanos P, Kuhn D, Rustem B (2012) A constraint sampling approach for multi-stage robust optimization. Automatica 48(3):459–471CrossRefGoogle Scholar
  109. Yanıkoğlu İ, Gorissen B, den Hertog D (2017) Adjustable robust optimization—a survey and tutorial. ResearchGateGoogle Scholar
  110. Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper Res Lett 41(5):457–461CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and EURO - The Association of European Operational Research Societies 2018

Authors and Affiliations

  1. 1.Fakultät für MathematikTU Dortmund UniversityDortmundGermany
  2. 2.Lehrstuhl C für Mathematik (Analysis)RWTH Aachen UniversityAachenGermany

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