Saddle point approximation approaches for two-stage robust optimization problems

  • Ning Zhang
  • Chang FangEmail author


This paper aims to present improvable and computable approximation approaches for solving the two-stage robust optimization problem, which arises from various applications such as optimal energy management and production planning. Based on sampling finite number scenarios of uncertainty, we can obtain a lower bound approximation and show that the corresponding solution is at least \({\varepsilon }\)-level feasible. Moreover, piecewise linear decision rules (PLDRs) are also introduced to improve the upper bound that obtained by the widely-used linear decision rule. Furthermore, we show that both the lower bound and upper bound approximation problems can be reformulated into solvable saddle point problems and consequently be solved by the mirror descent method.


Two-stage robust optimization Randomized approach Piecewise linear decision rule Saddle point problem Mirror descent algorithm 



We would like to thank Professor Xu Huan for his constructive suggestions on the idea of solving the two-stage robust optimization by using the randomized approach.


  1. 1.
    Agra, A., Christiansen, M., Figueiredo, R., Hvattum, L.M., Poss, M., Requejo, C.: The robust vehicle routing problem with time windows. Comput. Oper. Res. 40(3), 856–866 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    An, Y., Zeng, B.: Exploring the modeling capacity of two-stage robust optimization: variants of robust unit commitment model. IEEE Trans. Power Syst. 30(1), 109–122 (2015)CrossRefGoogle Scholar
  3. 3.
    Ardestani-Jaafari, A., Delage, E.: Robust optimization of sums of piecewise linear functions with application to inventory problems. Oper. Res. 64(2), 474–494 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ardestani-Jaafari, A., Delage, E.: The value of flexibility in robust location-transportation problems. Transp. Sci. 52(1), 189–209 (2017)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ben-Tal, A., Chung, B Do, Mandala, S .R., Yao, T.: Robust optimization for emergency logistics planning: risk mitigation in humanitarian relief supply chains. Transp. Res. Part B Methodol. 45(8), 1177–1189 (2011)CrossRefGoogle Scholar
  6. 6.
    Ben-Tal, A., El Housni, O., Goyal, V.: A tractable approach for designing piecewise affine policies in Two-stage adjustable robust optimization. Math. Program. (2019). CrossRefGoogle Scholar
  7. 7.
    Ben-Tal, A., Golany, B., Nemirovski, A., Vial, J.P.: Supplier-retailer flexible commitments contracts: a robust optimization approach. Manuf. Serv. Oper. Manag. 7(3), 248–271 (2005)CrossRefGoogle Scholar
  8. 8.
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99(2), 351–376 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bertsimas, D., Bidkhori, H.: On the performance of affine policies for two-stage adaptive optimization: a geometric perspective. Math. Program. 153(2), 577–594 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bertsimas, D., Goyal, V.: On the power of robust solutions in two-stage stochastic and adaptive optimization problems. Math. Oper. Res. 35(2), 284–305 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bertsimas, D., Goyal, V.: On the approximability of adjustable robust convex optimization under uncertainty. Math. Methods Oper. Res. 77(3), 323–343 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bertsimas, D., Goyal, V., Lu, B.Y.: A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization. Math. Program. 150(2), 281–319 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bertsimas, D., Iancu, D.A., Parrilo, P.A.: Optimality of affine policies in multistage robust optimization. Math. Oper. Res. 35(2), 363–394 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Calafiore, G., Campi, M.C.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102(1), 25–46 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Calafiore, G., Campi, M.C.: The scenario approach to robust control design. IEEE Trans. Automat. Control. 51(5), 742–753 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chou, M.C., Chua, G.A., Zheng, H.: On the performance of sparse process structures in partial postponement production systems. Oper. Res. 62(2), 348–365 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    De Farias, D.P., Van Roy, B.: On constraint sampling in the linear programming approach to approximate dynamic programming. Math. Oper. Res. 29(3), 462–478 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107(1–2), 37–61 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gabrel, V., Lacroix, M., Murat, C., Remli, N.: Robust location transportation problems under uncertain demands. Discrete Appl. Math. 164, 100–111 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Guslitser, E.: Uncertatinty-immunized solutions in linear programming. Master Thesis, Technion, Israeli Institute of Technology, IE&M faculty (2002)Google Scholar
  21. 21.
    Iancu, D.A., Sharma, M., Sviridenko, M.: Supermodularity and affine policies in dynamic robust optimization. Oper. Res. 61(4), 941–956 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Jane, J.Y., Zhang, J.: Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications. Math. Program. 139(1–2), 353–381 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Li, B., Wang, H., Yang, J., Guo, M., Qi, C.: A belief-rule-based inference method for aggregate production planning under uncertainty. Int. J. Prod. Res. 51(1), 83–105 (2013)CrossRefGoogle Scholar
  24. 24.
    Lorca, A., Sun, X.A.: Adaptive robust optimization with dynamic uncertainty sets for multi-period economic dispatch under significant wind. IEEE Trans. Power Syst. 30(4), 1702–1713 (2015)CrossRefGoogle Scholar
  25. 25.
    Mangasarian, O.L., Shiau, T.-H.: Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. SIAM J. Control Optim. 25(3), 583–595 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116(1–2), 369–396 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19(4), 1574–1609 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Paul, T.: On accelerated proximal gradient methods for convex-concave optimization. University of Washington, Seattle, Manuscript (2008)Google Scholar
  29. 29.
    Rahal, S., Papageorgiou, D.J., Li, Z.: Hybrid strategies using linear and piecewise-linear decision rules for multistage adaptive linear optimization. arXiv:1812.04522v1 (2018)
  30. 30.
    Robinson, S.M.: A characterization of stability in linear programming. Oper. Res. 25(3), 435–447 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Robinson, S.M.: Generalized equations and their solutions, Part I: basic theory. In: Point-to-Set Maps and Mathematical Programming, pp. 128–141. Springer, Berlin (1979)Google Scholar
  32. 32.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (2015)Google Scholar
  33. 33.
    See, S.M., Chuen-Teck, : Robust approximation to multiperiod inventory management. Oper. Res. 58(3), 583–594 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Simchi-Levi, D., Wang, H., Wei, Y.: Constraint generation for two-stage robust network flow problems. INFORMS J. Optim. 1(1), 49–70 (2018)CrossRefGoogle Scholar
  35. 35.
    Walkup, D.W., Wets, R.J.-B.: Stochastic programs with recourse. SIAM J. Appl. Math. 15(5), 1299–1314 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Wang, Q., Watson, J.-P., Guan, Y.: Two-stage robust optimization for N-k contingency-constrained unit commitment. IEEE Trans. Power Syst. 28(3), 2366–2375 (2013)CrossRefGoogle Scholar
  37. 37.
    Wets, R.J.-B.: Stochastic programs with fixed recourse: the equivalent deterministic program. SIAM Rev. 16(3), 309–339 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Xin, C., Zhang, Y.: Uncertain linear programs: extended affinely adjustable robust counterparts. Oper. Res. 57(6), 1469–1482 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Xu, G., Burer, S.: A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides. Comput. Optim. Appl. 70(1), 33–59 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Yanıkoğlu, İ., Gorissen, B.L., den Hertog, D.: A survey of adjustable robust optimization. Eur. J. Oper. Res. 277(3), 799–813 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Yuan, W., Wang, J., Qiu, F., Chen, C., Kang, C., Zeng, B.: Robust optimization-based resilient distribution network planning against natural disasters. IEEE Trans. Smart Grid 7(6), 2817–2826 (2016)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyDongguan University of TechnologyDongguanChina
  2. 2.School of Economics and ManagementAnhui Normal UniversityWuhu CityChina

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