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Robust combinatorial optimization under convex and discrete cost uncertainty

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EURO Journal on Computational Optimization

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.

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References

  • Adjiashvili D, Zenklusen R (2011) An s–t connection problem with adaptability. Discrete Appl Math 159(8):695–705

    Article  Google Scholar 

  • Adjiashvili D, Stiller S, Zenklusen R (2015) Bulk-robust combinatorial optimization. Math Program 149(1–2):361–390

    Article  Google Scholar 

  • 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

  • Adjiashvili D, Bindewald V, Michaels D (2017) Robust assignments with vulnerable nodes. Technical report. http://arxiv.org/abs/1703.06074

  • 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 862–873

  • Aissi H, Bazgan C, Vanderpooten D (2005b) Complexity of the min–max and min–max regret assignment problems. Oper Res Lett 33(6):634–640

    Article  Google Scholar 

  • Aissi H, Bazgan C, Vanderpooten D (2005c) Complexity of the min–max (regret) versions of cut problems. In: Algorithms and computation. Springer, Berlin, pp 789–798

  • 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 171–178

  • 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):427–438

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Armon A, Zwick U (2006) Multicriteria global minimum cuts. Algorithmica 46(1):15–26

    Article  Google Scholar 

  • Atamtürk A (2006) Strong formulations of robust mixed 0–1 programming. Math Program 108:235–250

    Article  Google Scholar 

  • Atamtürk A, Narayanan V (2008) Polymatroids and mean-risk minimization in discrete optimization. Oper Res Lett 36(5):618–622

    Article  Google Scholar 

  • Atamtürk A, Zhang M (2007) Two-stage robust network flow and design under demand uncertainty. Oper Res 55(4):662–673

    Article  Google Scholar 

  • Averbakh I, Lebedev V (2004) Interval data minmax regret network optimization problems. Discrete Appl Math 138(3):289–301

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Ayoub J, Poss M (2016) Decomposition for adjustable robust linear optimization subject to uncertainty polytope. Comput Manag Sci 13(2):219–239

    Article  Google Scholar 

  • 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 62–74

  • Baumann F, Buchheim C, Ilyina A (2015) A Lagrangean decomposition approach for robust combinatorial optimization. Technical report, Optimization online

  • Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805

    Article  Google Scholar 

  • Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13

    Article  Google Scholar 

  • Ben-Tal A, Nemirovski A (2002) Robust optimization-methodology and applications. Math Program 92(3):453–480

    Article  Google Scholar 

  • Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2):351–376

    Article  Google Scholar 

  • 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):248–271

    Article  Google Scholar 

  • Bertsimas D, Caramanis C (2010) Finite adaptability in multistage linear optimization. IEEE Trans Autom Control 55(12):2751–2766

    Article  Google Scholar 

  • Bertsimas D, Dunning I (2016) Multistage robust mixed-integer optimization with adaptive partitions. Oper Res 64(4):980–998

    Article  Google Scholar 

  • Bertsimas D, Georghiou A (2014) Binary decision rules for multistage adaptive mixed-integer optimization. Math Program 167:1–39

    Google Scholar 

  • Bertsimas D, Georghiou A (2015) Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Oper Res 63(3):610–627

    Article  Google Scholar 

  • Bertsimas D, Goyal V (2013) On the approximability of adjustable robust convex optimization under uncertainty. Math Methods Oper Res 77(3):323–343

    Article  Google Scholar 

  • Bertsimas D, Lubin IDM (2016) Reformulation versus cutting-planes for robust optimization. Comput Manag Sci 13(2):195–217

    Article  Google Scholar 

  • Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98(1–3):49–71

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Bertsimas D, Sim M (2004b) Robust discrete optimization under ellipsoidal uncertainty sets. Citeseer

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

    Article  Google Scholar 

  • Bertsimas D, Iancu DA, Parrilo PA (2010) Optimality of affine policies in multistage robust optimization. Math Oper Res 35(2):363–394

    Article  Google Scholar 

  • Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501

    Article  Google Scholar 

  • Beyer HG, Sendhoff B (2007) Robust optimization—a comprehensive survey. Comput Methods Appl Mech Eng 196(33):3190–3218

    Article  Google Scholar 

  • Billionnet A, Costa MC, Poirion PL (2014) 2-Stage robust MILP with continuous recourse variables. Discrete Appl Math 170:21–32

    Article  Google Scholar 

  • Buchheim C, Kurtz J (2016) Min–max–min robust combinatorial optimization subject to discrete uncertainty. Optimization online

  • Buchheim C, Kurtz J (2017) Min–max–min robust combinatorial optimization. Math Program 163(1):1–23

    Article  Google Scholar 

  • 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 online

  • Büsing C (2011) Recoverable robustness in combinatorial optimization. Ph.D. thesis, Technical University of Berlin

  • Büsing C (2012) Recoverable robust shortest path problems. Networks 59(1):181–189

    Article  Google Scholar 

  • Büsing C, D’Andreagiovanni F (2012) New results about multi-band uncertainty in robust optimization. In: International symposium on experimental algorithms. Springer, Berlin, pp 63–74

  • Büsing C, D’Andreagiovanni F (2013) Robust optimization under multi-band uncertainty—part I: theory. arXiv preprint arXiv:1301.2734

  • Büsing C, Koster A, Kutschka M (2011a) Recoverable robust knapsacks: \(\gamma \)-scenarios. Network optimization. Springer, Berlin, pp 583–588

    Google Scholar 

  • Büsing C, Koster AM, Kutschka M (2011b) Recoverable robust knapsacks: the discrete scenario case. Optim Lett 5(3):379–392

    Article  Google Scholar 

  • Calafiore GC (2008) Multi-period portfolio optimization with linear control policies. Automatica 44(10):2463–2473

    Article  Google Scholar 

  • Chang TJ, Meade N, Beasley J, Sharaiha Y (2000) Heuristics for cardinality constrained portfolio optimisation. Comput Oper Res 27:1271–1302

    Article  Google Scholar 

  • Chassein A, Goerigk M (2016) Min–max regret problems with ellipsoidal uncertainty sets. arXiv preprint arXiv:1606.01180

  • 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

  • Chen X, Zhang Y (2009) Uncertain linear programs: extended affinely adjustable robust counterparts. Oper Res 57(6):1469–1482

    Article  Google Scholar 

  • Claßen G, Koster AM, Schmeink A (2015) The multi-band robust knapsack problem—a dynamic programming approach. Discrete Optim 18:123–149

    Article  Google Scholar 

  • Cornuejols G, Tütüncü R (2006) Optimization methods in finance, vol 5. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Corporation I (2015) IBM ILOG CPLEX optimization studio: CPLEX user’s manual. https://www.ibm.com/us-en/marketplace/ibm-ilog-cplex

  • El Ghaoui L, Lebret H (1997) Robust solutions to least-squares problems with uncertain data. SIAM J Matrix Anal Appl 18(4):1035–1064

    Article  Google Scholar 

  • El Ghaoui L, Oustry F, Lebret H (1998) Robust solutions to uncertain semidefinite programs. SIAM J Optim 9(1):33–52

    Article  Google Scholar 

  • Feige U, Jain K, Mahdian M, Mirrokni V (2007) Robust combinatorial optimization with exponential scenarios. In: Integer programming and combinatorial optimization, pp 439–453

  • Fischetti M, Monaci M (2009) Light robustness. In: Ahuja RK, Möhring RH, Zaroliagis CD (eds) Robust and online large-scale optimization. Springer, Berlin, pp 61–84

    Chapter  Google Scholar 

  • Fischetti M, Monaci M (2012) Cutting plane versus compact formulations for uncertain (integer) linear programs. Math Program Comput 4:1–35

    Article  Google Scholar 

  • Fischetti M, Salvagnin D, Zanette A (2009) Fast approaches to improve the robustness of a railway timetable. Transp Sci 43(3):321–335

    Article  Google Scholar 

  • Gabrel V, Murat C, Thiele A (2014) Recent advances in robust optimization: an overview. Eur J Oper Res 235(3):471–483

    Article  Google Scholar 

  • Georghiou A, Wiesemann W, Kuhn D (2015) Generalized decision rule approximations for stochastic programming via liftings. Math Program 152(1–2):301–338

    Article  Google Scholar 

  • Gorissen BL, Yanıkoğlu İ, den Hertog D (2015) A practical guide to robust optimization. Omega 53:124–137

    Article  Google Scholar 

  • Grötschel M, Lovász L, Schrijver A (1993) Geometric algorithms and combinatorial optimization. Springer, Berlin

    Book  Google Scholar 

  • Gurobi Optimization I (2016) Gurobi optimizer reference manual. http://www.gurobi.com. Accessed 3 Sept 2018

  • Hanasusanto GA, Kuhn D, Wiesemann W (2015) K-Adaptability in two-stage robust binary programming. Oper Res 63(4):877–891

    Article  Google Scholar 

  • Hradovich M, Kasperski A, Zieliński P (2016) The robust recoverable spanning tree problem with interval costs is polynomially solvable. arXiv preprint arXiv:1602.07422

  • Iancu DA (2010) Adaptive robust optimization with applications in inventory and revenue management. Ph.D. thesis, Massachusetts Institute of Technology

  • Iancu DA, Sharma M, Sviridenko M (2013) Supermodularity and affine policies in dynamic robust optimization. Oper Res 61(4):941–956

    Article  Google Scholar 

  • Ilyina A (2017) Combinatorial optimization under ellipsoidal uncertainty. Ph.D. thesis, TU Dortmund University

  • 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

    Article  Google Scholar 

  • Kasperski A, Zieliński P (2011) On the approximability of robust spanning tree problems. Theor Comput Sci 412(4–5):365–374

    Article  Google Scholar 

  • Kasperski A, Zieliński P (2015) Robust recoverable and two-stage selection problems. arXiv preprint arXiv:1505.06893

  • 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 113–143

    Google Scholar 

  • Kasperski A, Zieliński P (2017) Robust two-stage network problems. In: Operations research proceedings 2015. Springer, Berlin, pp 35–40

  • Kasperski A, Kurpisz A, Zieliński P (2014) Recoverable robust combinatorial optimization problems. In: Operations research proceedings 2012. Springer, Berlin, pp 147–153

  • Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, Berlin

    Book  Google Scholar 

  • Khandekar R, Kortsarz G, Mirrokni V, Salavatipour M (2008) Two-stage robust network design with exponential scenarios. Algorithms ESA 2008:589–600

    Google Scholar 

  • Kouvelis P, Yu G (1996) Robust discrete optimization and its applications. Springer, Berlin

    Google Scholar 

  • Kuhn D, Wiesemann W, Georghiou A (2011) Primal and dual linear decision rules in stochastic and robust optimization. Math Program 130(1):177–209

    Article  Google Scholar 

  • Kurtz J (2016) Min–max–min robust combinatorial optimization. Ph.D. thesis, TU Dortmund University

  • Lappas NH, Gounaris CE (2018) Robust optimization for decision-making under endogenous uncertainty. Comput Chem Eng 111:252–266

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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):10567–10603

    Article  Google Scholar 

  • 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 1–27

    Google Scholar 

  • Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91

    Google Scholar 

  • Minoux M (2011) On 2-stage robust LP with RHS uncertainty: complexity results and applications. J Glob Optim 49(3):521–537

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Mutapcic A, Boyd S (2009) Cutting-set methods for robust convex optimization with pessimizing oracles. Optim Methods Softw 24(3):381–406

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Nasrabadi E, Orlin JB (2013) Robust optimization with incremental recourse. arXiv preprint arXiv:1312.4075

  • Nikolova E (2010a) Approximation algorithms for offline risk-averse combinatorial optimization. Technical report

  • 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 338–351

    Chapter  Google Scholar 

  • Nohadani O, Sharma K (2016) Optimization under decision-dependent uncertainty. arXiv preprint arXiv:1611.07992

  • Park KC, Lee KS (2007) A note on robust combinatorial optimization problem. Manag Sci Financ Eng 13(1):115–119

    Google Scholar 

  • Pessoa AA, Poss M (2015) Robust network design with uncertain outsourcing cost. INFORMS J Comput 27(3):507–524

    Article  Google Scholar 

  • Poss M (2013) Robust combinatorial optimization with variable budgeted uncertainty. 4OR 11(1):75–92

    Article  Google Scholar 

  • Poss M (2017) Robust combinatorial optimization with knapsack uncertainty. Discrete Optim 27:88–102

    Article  Google Scholar 

  • Postek K, den Hertog D (2016) Multistage adjustable robust mixed-integer optimization via iterative splitting of the uncertainty set. INFORMS J Comput 28(3):553–574

    Article  Google Scholar 

  • Saito H, Murota K (2007) Benders decomposition approach to robust mixed integer programming. Pac J Optim 3:99–112

    Google Scholar 

  • Schöbel A (2014) Generalized light robustness and the trade-off between robustness and nominal quality. Math Methods Oper Res 80:1–31

    Article  Google Scholar 

  • Shapiro A (2011) A dynamic programming approach to adjustable robust optimization. Oper Res Lett 39(2):83–87

    Article  Google Scholar 

  • Sim M (2004) Robust optimization. Ph.D. thesis, Massachusetts Institute of Technology

  • Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper Res 21(5):1154–1157

    Article  Google Scholar 

  • Subramanyam A, Gounaris CE, Wiesemann W (2017) K-Adaptability in two-stage mixed-integer robust optimization. arXiv preprint arXiv:1706.07097

  • Vayanos P, Kuhn D, Rustem B (2012) A constraint sampling approach for multi-stage robust optimization. Automatica 48(3):459–471

    Article  Google Scholar 

  • Yanıkoğlu İ, Gorissen B, den Hertog D (2017) Adjustable robust optimization—a survey and tutorial. ResearchGate

  • Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper Res Lett 41(5):457–461

    Article  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, TU Dortmund University, Vogelpothsweg 87, 44227, Dortmund, Germany

    Christoph Buchheim

  2. Lehrstuhl C für Mathematik (Analysis), RWTH Aachen University, Pontdriesch 10, 52062, Aachen, Germany

    Jannis Kurtz

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  1. Christoph Buchheim
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  2. Jannis Kurtz
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Correspondence to Jannis Kurtz.

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The second author has been supported by the German Research Foundation (DFG) within the Research Training Group 1855 and under Grant BU 2313/2.

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Buchheim, C., Kurtz, J. Robust combinatorial optimization under convex and discrete cost uncertainty. EURO J Comput Optim 6, 211–238 (2018). https://doi.org/10.1007/s13675-018-0103-0

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