Abstract
In this paper, we study uncertainty set construction for robust optimization using various polyhedral norms. We first introduce the classical symmetric polyhedral-norms induced uncertainty sets and the corresponding robust counterparts of a linear uncertain constraint. Then, we introduce a novel method for asymmetric uncertainty set construction based on the distributional information of the uncertain parameters. Deterministic robust counterpart formulations for both types of uncertainty sets are derived for a general linear uncertain constraint. We further derive the robust counterpart of a linear uncertain constraint where the uncertain parameters belong to (i) an intersection of two symmetric uncertainty sets and (ii) an intersection of asymmetric and symmetric uncertainty sets. Using a numerical example and a reactor design problem, we demonstrate that appropriate uncertainty set construction reduces solution conservativeness. We also highlight the significance of integrating the data and distributional information of uncertain parameters in terms of safeguarding feasibility alongside improving the solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88(3):411–424
Ben-Tal A, Ghaoui LE, Nemirovski A (2000) Robustness, Springer, pp 139–162. Handbook of semidefinite programming
Ben-Tal A, Nemirovski A, Roos C (2002) Robust solutions of uncertain quadratic and conic-quadratic problems. SIAM J Optim 13(2):535–560
Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2):351–376
Ben-Tal A, Ghaoui LE, Nemirovski A (2009) Robust Optimization, vol 28. Princeton University Press, Princeton
Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52(1):35–53
Bertsimas D, Pachamanova D, Sim M (2004) Robust linear optimization under general norms. Oper Res Lett 32(6):510–516
Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM
Boyd S, Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge
Chen X, Sim M, Sun P (2007) A robust optimization perspective on stochastic programming. Oper Res 55(6):1058–1071
Dai X, Wang X, He R, Du W, Zhong W, Zhao L, Qian F (2020) Data-driven robust optimization for crude oil blending under uncertainty. Comput Chem Eng 136:106595
Garuba F, Goerigk M, Jacko P (2020) A comparison of data-driven uncertainty sets for robust network design. arXiv preprint arXiv:200310507
Gotoh J, Uryasev S (2016) Two pairs of families of polyhedral norms versus \(\ell _p \)-norms: proximity and applications in optimization. Math Program 156(1–2):391–431
Guzman YA, Matthews LR, Floudas CA (2016) New a priori and a posteriori probabilistic bounds for robust counterpart optimization: I. unknown probability distributions. Comput Chem Eng 84:568–598
Guzman YA, Matthews LR, Floudas CA (2017a) New a priori and a posteriori probabilistic bounds for robust counterpart optimization: Ii. a priori bounds for known symmetric and asymmetric probability distributions. Comput Chem Eng 101:279–311
Guzman YA, Matthews LR, Floudas CA (2017b) New a priori and a posteriori probabilistic bounds for robust counterpart optimization: Iii. exact and near-exact a posteriori expressions for known probability distributions. Comput Chem Eng 103:116–143
Kusunoki Y, Tatsumi K (2019) Scalarization for approximate multiobjective multiclass support vector machine using the large-\( k \) norm. In: 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019), Atlantis Press
Li C, Zhao J, Zheng T, Litvinov E (2016) Data-driven uncertainty sets: Robust optimization with temporally and spatially correlated data. In: 2016 IEEE Power and Energy Society General Meeting (PESGM), IEEE, pp 1–5
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
Li Z, Tang Q, Floudas CA (2012) A comparative theoretical and computational study on robust counterpart optimization: Ii. probabilistic guarantees on constraint satisfaction. Ind Eng Chem Res 51(19):6769–6788
Matthews LR, Guzman YA, Floudas CA (2018) Generalized robust counterparts for constraints with bounded and unbounded uncertain parameters. Computers & Chemical Engineering 116:451–467, multi-scale Systems Engineering – in memory & honor of Professor C.A. Floudas
Ning C, You F (2017) A data-driven multistage adaptive robust optimization framework for planning and scheduling under uncertainty. AIChE J 63(10):4343–4369
Ning C, You F (2018a) Data-driven decision making under uncertainty integrating robust optimization with principal component analysis and kernel smoothing methods. Comput Chem Eng 112:190–210
Ning C, You F (2018b) Data-driven stochastic robust optimization: General computational framework and algorithm leveraging machine learning for optimization under uncertainty in the big data era. Comput Chem Eng 111:115–133
Ning C, You F (2019a) Data-driven adaptive robust unit commitment under wind power uncertainty: A bayesian nonparametric approach. IEEE Trans Power Syst 34(3):2409–2418
Ning C, You F (2019b) Optimization under uncertainty in the era of big data and deep learning: When machine learning meets mathematical programming. Comput Chem Eng 125:434–448
Shang C, Chen WH, You F (2019) Robust constrained model predictive control of irrigation systems based on data-driven uncertainty set constructions. In: 2019 American Control Conference (ACC), IEEE, pp 1–6
Shen F, Zhao L, Du W, Zhong W, Qian F (2020) Large-scale industrial energy systems optimization under uncertainty: A data-driven robust optimization approach. Appl Energy 259:114199
Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper Res 21(5):1154–1157
Yanıkoğlu I, Gorissen BL, den Hertog D (2019) A survey of adjustable robust optimization. Euro J Oper Res 277(3):799–813
Yuan Y, Li Z, Huang B (2016) Robust optimization under correlated uncertainty: Formulations and computational study. Comput Chem Eng 85:58–71
Zhang Y, Jin X, Feng Y, Rong G (2018) Data-driven robust optimization under correlated uncertainty: A case study of production scheduling in ethylene plant. Comput Chem Eng 109:48–67
Zhao L, Zhong W, Du W (2019) Data-driven robust optimization for steam systems in ethylene plants under uncertainty. Processes 7(10):744
Acknowledgements
The authors gratefully acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rahal, S., Li, Z. Norm induced polyhedral uncertainty sets for robust linear optimization. Optim Eng 23, 1765–1801 (2022). https://doi.org/10.1007/s11081-021-09659-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11081-021-09659-3