Skip to main content

Advertisement

SpringerLink
Log in

A survey of robust optimization based machine learning with special reference to support vector machines

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

This paper gives an overview of developments in the field of robust optimization in machine learning (ML) in general and Support Vector Machine (SVM)/Support Vector Regression (SVR) models in particular. This survey comprises of researches in which robustness is sought against uncertainty. This uncertainty is in the values of parameters of the given model or it can be in the data. In this work, we have discussed how robust optimization has entered in the field of machine learning. Here, we investigate the contribution of various researchers in dealing with different types of uncertainties arising in the problem of maximizing or minimizing the objective function. We deal with the variants of SVM/SVR in more detail, although we have also covered supervised, unsupervised ML and various other aspects of ML. Also, we present an extensive study of research carried out in the applications of robust optimization in other areas like energy and power systems, networking, transportation etc. as well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Abe S (2005) Support vector machines for pattern classification, vol 2. Springer, Berlin

    MATH  Google Scholar 

  2. Agra A, Christiansen M, Figueiredo R, Hvattum LM, Poss M, Requejo C (2013) The robust vehicle routing problem with time windows. Comput Oper Res 40(3):856–866

    Article  MathSciNet  MATH  Google Scholar 

  3. Agra A, Christiansen M, Hvattum LM, Rodrigues F (2018) Robust optimization for a maritime inventory routing problem. Transportation Science

  4. Ahmed MB, Mansour FZ, Haouari M (2017) A two-level optimization approach for robust aircraft routing and retiming. Comput Ind Eng 112:586–594

    Article  Google Scholar 

  5. Ahmed MB, Mansour FZ, Haouari M (2018) Robust integrated maintenance aircraft routing and crew pairing. J Air Transport Manag 73:15–31

    Article  Google Scholar 

  6. An Y, Zeng B, Zhang Y, Zhao L (2014) Reliable p-median facility location problem: two-stage robust models and algorithms. Transp Res Part B Methodol 64:54–72

    Article  Google Scholar 

  7. Angulo C, Anguita D, Gonzalez-Abril L, Ortega JA (2008) Support vector machines for interval discriminant analysis. Neurocomputing 71(7–9):1220–1229

    Article  Google Scholar 

  8. Ardestani-Jaafari A (2016) Linearized robust counterparts with applications in location and inventory management problems. HEC Montreal (Canada)

  9. Astorino A, Bomze I, Fuduli A, Gaudioso M (2017) Robust spherical separation. Optimization 66(6):925–938

    Article  MathSciNet  MATH  Google Scholar 

  10. Babaei M, Alkhatib A, Pan I (2015) Robust optimization of subsurface flow using polynomial chaos and response surface surrogates. Comput Geosci 19(5):979–998

    Article  MathSciNet  MATH  Google Scholar 

  11. Bai MR, Chen CC (2014) Regularization using monte carlo simulation to make optimal beamformers robust to system perturbations. J Acoust Soc Am 135(5):2808–2820

    Article  Google Scholar 

  12. Bandi C, Bertsimas D, Youssef N (2018) Robust transient analysis of multi-server queueing systems and feed-forward networks. Queueing Syst 201:1–63

    MathSciNet  MATH  Google Scholar 

  13. Baralis E, Cagliero L (2014) Rib: a robust itemset-based bayesian approach to classification. Knowl-Based Syst 71:366–375

    Article  Google Scholar 

  14. Baymani M, Salehi-M N, Mansoori A (2018) Applying norm concepts for solving interval support vector machine. Neurocomputing 311:41–50

    Article  Google Scholar 

  15. Beland JJ, Nair PB (2017) Bayesian optimization under uncertainty. In: NIPS BayesOpt 2017 workshop

  16. Bellazzi R, Demichelis F, Piergiorgi P, Magni P (2006) Hierarchical naive Bayes classifiers for uncertain data. In: DIS Technical Report, Semantic Scholar

  17. Ben-Hur A, Weston J (2010) A user’s guide to support vector machines. Data mining techniques for the life sciences. Springer, Berlin, pp 223–239

    Chapter  Google Scholar 

  18. Ben-Tal A, Nemirovski A (2008) Selected topics in robust convex optimization. Math Program 112(1):125–158

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton

    Book  MATH  Google Scholar 

  21. Ben-Tal A, Bhadra S, Bhattacharyya C, Nath JS (2011) Chance constrained uncertain classification via robust optimization. Math Program 127(1):145–173

    Article  MathSciNet  MATH  Google Scholar 

  22. Ben-Tal A, Bhadra S, Bhattacharyya C, Nemirovski A (2012) Efficient methods for robust classification under uncertainty in kernel matrices. J Mach Learn Res 13((Oct)):2923–2954

    MathSciNet  MATH  Google Scholar 

  23. Ben-Tal A, Den Hertog D, Vial JP (2015a) Deriving robust counterparts of nonlinear uncertain inequalities. Math Program 149(1–2):265–299

    Article  MathSciNet  MATH  Google Scholar 

  24. Ben-Tal A, Hazan E, Koren T, Mannor S (2015b) Oracle-based robust optimization via online learning. Oper Res 63(3):628–638

    Article  MathSciNet  MATH  Google Scholar 

  25. Ben-Tal A, Brekelmans R, Den Hertog D, Vial JP (2017) Globalized robust optimization for nonlinear uncertain inequalities. INFORMS J Comput 29(2):350–366

    Article  MathSciNet  MATH  Google Scholar 

  26. Bertsimas D, Copenhaver MS (2018) Characterization of the equivalence of robustification and regularization in linear and matrix regression. Eur J Oper Res 270(3):931–942

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Bertsimas D, Nohadani O, Teo KM (2010a) Nonconvex robust optimization for problems with constraints. INFORMS J Comput 22(1):44–58

    Article  MathSciNet  MATH  Google Scholar 

  29. Bertsimas D, Nohadani O, Teo KM (2010b) Robust optimization for unconstrained simulation-based problems. Oper Res 58(1):161–178

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Bertsimas D, Goyal V, Lu BY (2015a) A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization. Math Program 150(2):281–319

    Article  MathSciNet  MATH  Google Scholar 

  32. Bertsimas D, Nasrabadi E, Paschalidis IC (2015b) Robust fluid processing networks. IEEE Trans Autom Control 60(3):715–728

    Article  MathSciNet  MATH  Google Scholar 

  33. Bertsimas D, Gupta V, Kallus N (2018) Data-driven robust optimization. Math Program 167(2):235–292

    Article  MathSciNet  MATH  Google Scholar 

  34. Bhadra S (2012) Learning robust support vector machine classifiers with uncertain observations. PhD thesis, Citeseer

  35. Bhadra S, Bhattacharya S, Bhattacharyya C, Ben-Tal A (2010) Robust formulations for handling uncertainty in kernel matrices. In: Proceedings of the 27th international conference on machine learning (ICML-10), pp 71–78

  36. Bhattacharyya C, Grate L, Jordan MI, Ghaoui LE, Mian IS (2004) Robust sparse hyperplane classifiers: application to uncertain molecular profiling data. J Comput Biol 11(6):1073–1089

    Article  Google Scholar 

  37. Bi J, Zhang T (2005) Support vector classification with input data uncertainty. In: Advances in neural information processing systems, pp 161–168

  38. Bogunovic I, Scarlett J, Jegelka S, Cevher V (2018) Adversarially robust optimization with Gaussian processes. In: Advances in neural information processing systems, pp 5763–5773

  39. Burges CJ (1998) A tutorial on support vector machines for pattern recognition. Data Min Knowl Disc 2(2):121–167

    Article  Google Scholar 

  40. Büttner S, Krumke SO (2016) Robust optimization for routing problems on trees. TOP 24(2):338–359

    Article  MathSciNet  MATH  Google Scholar 

  41. Calvo B, Larrañaga P, Lozano JA (2007) Learning bayesian classifiers from positive and unlabeled examples. Pattern Recogn Lett 28(16):2375–2384

    Article  Google Scholar 

  42. Cao Q, Fu X, Guo Y (2017) Fuzzy chance constrained twin support vector machine for uncertain classification. In: International conference on management science and engineering management, Springer, Berlin, pp 1508–1521

  43. Caramanis C, Mannor S, Xu H (2012) 14 robust optimization in machine learning. Optim Mach Learn 20:369

    Google Scholar 

  44. Chen D, Kuehn V (2017) Robust resource allocation and clustering formulation for multicast c-ran with impaired CSI. In: 2017 IEEE international conference on communications (ICC), IEEE, pp 1–6

  45. Chen J, Ji G (2010) Weighted least squares twin support vector machines for pattern classification. In: 2010 the 2nd international conference on computer and automation engineering (ICCAE), IEEE, vol 2, pp 242–246

  46. Chen X, Sim M, Sun P (2007) A robust optimization perspective on stochastic programming. Oper Res 55(6):1058–1071

    Article  MathSciNet  MATH  Google Scholar 

  47. Couellan N, Wang W (2017) Uncertainty-safe large scale support vector machines. Comput Stat Data Anal 109:215–230

    Article  MathSciNet  MATH  Google Scholar 

  48. Crémilleux B, Robert C (1997) A theoretical framework for decision trees in uncertain domains: application to medical data sets. In: Conference on artificial intelligence in medicine in Europe. Springer, pp 143–156

  49. Denoeux T (2013) Maximum likelihood estimation from uncertain data in the belief function framework. IEEE Trans Knowl Data Eng 25(1):119–130

    Article  Google Scholar 

  50. Du P, Baldick R, Tuohy A (2017) Integration of large-scale renewable energy into bulk power systems. Springer, Berlin

    Book  Google Scholar 

  51. Duchi J, Glynn P, Namkoong H (2016) Statistics of robust optimization: a generalized empirical likelihood approach. arXiv:161003425 (arXiv preprint)

  52. 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  MathSciNet  MATH  Google Scholar 

  53. El Ghaoui L, Lanckriet GRG, Natsoulis G et al (2003) Robust classification with interval data

  54. Elliott SJ, Cheer J, Choi JW, Kim Y (2012) Robustness and regularization of personal audio systems. IEEE Trans Audio Speech Lang Process 20(7):2123–2133

    Article  Google Scholar 

  55. Falck T, Suykens JA, De Moor B (2009) Robustness analysis for least squares kernel based regression: an optimization approach. In: Decision and control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE conference on, IEEE, pp 6774–6779

  56. Fan N, Sadeghi E, Pardalos PM (2014) Robust support vector machines with polyhedral uncertainty of the input data. In: International conference on learning and intelligent optimization. Springer, pp 291–305

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

    Article  MathSciNet  MATH  Google Scholar 

  58. Gal Y, Ghahramani Z (2016) Dropout as a Bayesian approximation: representing model uncertainty in deep learning. In: International conference on machine learning, pp 1050–1059

  59. Gan H, Zhang Y, Song Q (2017) Bayesian belief network for positive unlabeled learning with uncertainty. Pattern Recogn Lett 90:28–35

    Article  Google Scholar 

  60. Ge J, Xia Y, Nadungodage C (2010) UNN: a neural network for uncertain data classification. In: Pacific-Asia conference on knowledge discovery and data mining. Springer, pp 449–460

  61. Geletu A, Klöppel M, Hoffmann A, Li P (2015) A tractable approximation of non-convex chance constrained optimization with non-gaussian uncertainties. Eng Optim 47(4):495–520

    Article  MathSciNet  Google Scholar 

  62. Geng X, Gupta S, Xie L (2018) Robust look-ahead three-phase balancing of uncertain distribution loads. arXiv:181000425 (arXiv preprint)

  63. Ghate A (2016) Robust optimization in countably infinite linear programs. Optim Lett 10(4):847–863

    Article  MathSciNet  MATH  Google Scholar 

  64. Ghelichi Z, Tajik J, Pishvaee MS (2018) A novel robust optimization approach for an integrated municipal water distribution system design under uncertainty: a case study of mashhad. Comput Chem Eng 110:13–34

    Article  Google Scholar 

  65. Goh J, Sim M (2010) Distributionally robust optimization and its tractable approximations. Oper Res 58 (Part–1)(4–1):902–917

    Article  MathSciNet  MATH  Google Scholar 

  66. Golmohamadi H, Keypour R (2017) Application of robust optimization approach to determine optimal retail electricity price in presence of intermittent and conventional distributed generation considering demand response. J Control Autom Electr Syst 28(5):664–678

    Article  Google Scholar 

  67. Golmohamadi H, Keypour R, Bak-Jensen B, Pillai JR (2019) Optimization of household energy consumption towards day-ahead retail electricity price in home energy management systems. Sustain Cities Soc 20:101468

    Article  Google Scholar 

  68. Golovin D, Goyal V, Polishchuk V, Ravi R, Sysikaski M (2015) Improved approximations for two-stage min-cut and shortest path problems under uncertainty. Math Program 149(1–2):167–194

    Article  MathSciNet  MATH  Google Scholar 

  69. González-Arribas D, Soler M, Sanjurjo-Rivo M, Kamgarpour M, Simarro J (2019) Robust aircraft trajectory planning under uncertain convective environments with optimal control and rapidly developing thunderstorms. Aerosp Sci Technol

  70. Gorissen B, Ben-Tal A, Blanc H, Den Hertog D (2012) A new method for deriving robust and globalized robust solutions of uncertain linear conic optimization problems having general convex uncertainty sets. SSRN

  71. Gorissen BL, h lu, den Hertog D (2015) A practical guide to robust optimization. Omega 53:124–137

    Article  Google Scholar 

  72. Grossmann IE, Apap RM, Calfa BA, Garcia-Herreros P, Zhang Q (2015) Recent advances in mathematical programming techniques for the optimization of process systems under uncertainty. In: Computer aided chemical engineering, vol 37, Elsevier, pp 1–14

  73. Gupta A, Nagarajan V, Ravi R (2010) Thresholded covering algorithms for robust and max-min optimization. In: International colloquium on automata, languages, and programming. Springer, pp 262–274

  74. Han K, Liu H, Gayah VV, Friesz TL, Yao T (2016) A robust optimization approach for dynamic traffic signal control with emission considerations. Transp Res Part C Emerg Technol 70:3–26

    Article  Google Scholar 

  75. Han R, Cao Q (2017) Fuzzy chance constrained least squares twin support vector machine for uncertain classification. J Intell Fuzzy Syst 33(5):3041–3049

    Article  Google Scholar 

  76. He J, Zhang Y, Li X, Shi P (2012) Learning naive bayes classifiers from positive and unlabelled examples with uncertainty. Int J Syst Sci 43(10):1805–1825

    Article  MathSciNet  MATH  Google Scholar 

  77. Hernandez I, Ramirez-Marquez JE, Rainwater C, Pohl E, Medal H (2014) Robust facility location: hedging against failures. Reliabil Eng Syst Saf 123:73–80

    Article  Google Scholar 

  78. Ho-Nguyen N, Kılınç-Karzan F (2016) Accelerating optimization under uncertainty via online convex optimization. Technical report. http://www.optimization-online.org/DB_HTML

  79. Ho-Nguyen N, Kılınç-Karzan F (2018) Online first-order framework for robust convex optimization. Oper Res

  80. Hora J, Dias TG, Camanho A (2015) Improving the robustness of bus schedules using an optimization model. Operations research and big data. Springer, Berlin, pp 79–87

    Chapter  Google Scholar 

  81. Huang G, Song S, Wu C, You K (2012) Robust support vector regression for uncertain input and output data. IEEE Trans Neural Netw Learn Syst 23(11):1690–1700

    Article  Google Scholar 

  82. Jalilvand-Nejad A, Shafaei R, Shahriari H (2016) Robust optimization under correlated polyhedral uncertainty set. Comput Ind Eng 92:82–94

    Article  Google Scholar 

  83. Kaddour M (2016) Robust routing and link scheduling in cognitive radio networks under channel uncertainty and demand fluctuations. In: Consumer communications and networking conference (CCNC), 2016 13th IEEE annual, IEEE, pp 212–217

  84. Karasik R, Simeone O, Shitz SS (2013) Robust uplink communications over fading channels with variable backhaul connectivity. IEEE Trans Wirel Commun 12(11):5788–5799

    Article  Google Scholar 

  85. Khandekar R, Kortsarz G, Mirrokni V, Salavatipour MR (2013) Two-stage robust network design with exponential scenarios. Algorithmica 65(2):391–408

    Article  MathSciNet  MATH  Google Scholar 

  86. Khemchandani R, Chandra S et al (2007) Twin support vector machines for pattern classification. IEEE Trans Pattern Anal Mach Intell 29(5):905–910

    Article  MATH  Google Scholar 

  87. Korolko N, Sahinoglu Z (2017) Robust optimization of ev charging schedules in unregulated electricity markets. IEEE Trans Smart Grid 8(1):149–157

    Article  Google Scholar 

  88. Krumke SO, Schmidt E, Streicher M (2019) Robust multicovers with budgeted uncertainty. Eur J Oper Res 274(3):845–857

    Article  MathSciNet  MATH  Google Scholar 

  89. Langley P, Simon HA (1995) Applications of machine learning and rule induction. Commun ACM 38(11):54–64

    Article  Google Scholar 

  90. Le Thi HA, Vo XT, Dinh TP (2013) Robust feature selection for SVMS under uncertain data. In: Industrial conference on data mining. Springer, pp 151–165

  91. Lei C, Lin WH, Miao L (2016) A two-stage robust optimization approach for the mobile facility fleet sizing and routing problem under uncertainty. Comput Oper Res 67:75–89

    Article  MathSciNet  MATH  Google Scholar 

  92. Li G, Ma AKC, Pong TK (2014) Robust least square semidefinite programming with applications. Comput Optim Appl 58(2):347–379

    Article  MathSciNet  MATH  Google Scholar 

  93. Li H, Pan D et al (2018) Optimizing fleet composition and size under uncertainty in urban transit systems

  94. Li WWL, Shen Y, Zhang YJ, Win MZ (2013) Robust power allocation for energy-efficient location-aware networks. IEEE/ACM Trans Netw 21(6):1918–1930

    Article  Google Scholar 

  95. Li Z, Wu W, Zhang B, Wang B (2015) Adjustable robust real-time power dispatch with large-scale wind power integration. IEEE Trans Sustain Energy 6(2):357–368

    Article  Google Scholar 

  96. Liang Z, Feng Y, Zhang X, Wu T, Chaovalitwongse WA (2015) Robust weekly aircraft maintenance routing problem and the extension to the tail assignment problem. Transp Res Part B Methodol 78:238–259

    Article  Google Scholar 

  97. Lieb AM (2016) Modeling and optimization of transients in water distribution networks with intermittent supply. PhD thesis, UC Berkeley

  98. Linhares A, Swamy C (2019) Approximation algorithms for distributionally robust stochastic optimization with black-box distributions. arXiv:190407381 (arXiv preprint)

  99. Liu D, Shi Y, Tian Y (2015a) Ramp loss nonparallel support vector machine for pattern classification. Knowl-Based Syst 85:224–233

    Article  Google Scholar 

  100. Liu H, Han K, Gayah VV, Friesz TL, Yao T (2015b) Data-driven linear decision rule approach for distributionally robust optimization of on-line signal control. Transp Res Part C Emerg Technol 59:260–277

    Article  Google Scholar 

  101. Liu Y, Zhang B, Chen B, Yang Y (2016) Robust solutions to fuzzy one-class support vector machine. Pattern Recogn Lett 71:73–77

    Article  Google Scholar 

  102. Liu Z, Li S, Ma K, Guan X, Li X (2017) Robust power allocation based on hierarchical game with consideration of different user requirements in two-tier femtocell networks. Comput Netw 122:179–190

    Article  Google Scholar 

  103. Liu Zg, Pan Q, Dezert J, Mercier G, Liu Y (2014) Fuzzy-belief k-nearest neighbor classifier for uncertain data. In: information fusion (FUSION), 2014 17th international conference on, IEEE, pp 1–8

  104. López J, Maldonado S (2018) Robust twin support vector regression via second-order cone programming. Knowl-Based Syst 152:83–93

    Article  Google Scholar 

  105. López J, Maldonado S, Carrasco M (2017) A robust formulation for twin multiclass support vector machine. Appl Intell 47(4):1031–1043

    Article  Google Scholar 

  106. Lou CX, Dai W (2012) Robust supply chain services system through optimization modeling for enterprises. In: Network-based information systems (NBiS), 2012 15th international conference on, IEEE, pp 518–523

  107. Lv Y (2014) Gan Z (2014) Robust \(\varepsilon \)-support vector regression. Math Prob Eng

  108. Mahmoudzadeh H, Lee J, Chan TC, Purdie TG (2015) Robust optimization methods for cardiac sparing in tangential breast imrt. Med Phys 42(5):2212–2222

    Article  Google Scholar 

  109. Maldonado S, López J, Carrasco M (2016) A second-order cone programming formulation for twin support vector machines. Appl Intell 45(2):265–276

    Article  Google Scholar 

  110. Mani M, Sing AK, Orshansky M (2006) Joint design-time and post-silicon minimization of parametric yield loss using adjustable robust optimization. In: Proceedings of the 2006 IEEE/ACM international conference on Computer-aided design, ACM, pp 19–26

  111. Marla L, Vaze V, Barnhart C (2018) Robust optimization: lessons learned from aircraft routing. Comput Oper Res 98:165–184

    Article  MathSciNet  MATH  Google Scholar 

  112. Matke C, Bienstock D, Munoz G, Yang S, Kleinhans D, Sager S (2016) Robust optimization of power network operation: storage devices and the role of forecast errors in renewable energies. In: International workshop on complex networks and their applications. Springer, pp 809–820

  113. Menickelly M, Wild SM (2017) Derivative-free robust optimization by outer approximations. Preprint ANL/MCS-P9004-1017 (Argonne National Laboratory, Mathematics and Computer Science Division, 2017) Google Scholar

  114. Moazeni S, Coleman TF, Li Y (2013) Regularized robust optimization: the optimal portfolio execution case. Comput Optim Appl 55(2):341–377

    Article  MathSciNet  MATH  Google Scholar 

  115. Moreira A, Street A, Arroyo JM (2015a) An adjustable robust optimization approach for contingency-constrained transmission expansion planning. IEEE Trans Power Syst 30(4):2013–2022

    Article  Google Scholar 

  116. Moreira A, Street A, Arroyo JM (2015b) Energy and reserve scheduling under correlated nodal demand uncertainty: an adjustable robust optimization approach. Int J Electr Power Energy Syst 72:91–98

    Article  Google Scholar 

  117. Mozafari K, Nasiri JA, Charkari NM, Jalili S (2011) Hierarchical least square twin support vector machines based framework for human action recognition. In: 2011 7th Iranian conference on machine vision and image processing, IEEE, pp 1–5

  118. Musa AB (2014) Logistic regression classification for uncertain data. Res J Math Stat Sci 2320:6047

    Google Scholar 

  119. Namkoong H, Duchi JC (2016) Stochastic gradient methods for distributionally robust optimization with f-divergences. In: Advances in neural information processing systems, pp 2208–2216

  120. Ng TS, Sy C (2014) An affine adjustable robust model for generation and transmission network planning. Int J Electr Power Energy Syst 60:141–152

    Article  Google Scholar 

  121. Ning C, You F (2017) Leveraging big data for adaptive robust optimization of scheduling under uncertainty. In: 2017 American control conference (ACC), IEEE, pp 3783–3788

  122. Nisar MD, Utschick W (2011) Minimax robust a priori information aware channel equalization. IEEE Trans Signal Process 59(4):1734–1745

    Article  MathSciNet  MATH  Google Scholar 

  123. Nojavan S, Ghesmati H, Zare K (2016) Robust optimal offering strategy of large consumer using IGDT considering demand response programs. Electr Power Syst Res 130:46–58

    Article  Google Scholar 

  124. Oros A, Topa M, Neag M, Rafaila M, Pelz G (2012) Application-oriented robustness optimization based on metamodels. In: Design and technology in electronic packaging (SIITME), 2012 IEEE 18th international symposium for, IEEE, pp 37–41

  125. O’Hanley JR, Church RL (2011) Designing robust coverage networks to hedge against worst-case facility losses. Eur J Oper Res 209(1):23–36

    Article  MathSciNet  MATH  Google Scholar 

  126. Park SH, Simeone O, Sahin O, Shamai S (2012) Robust distributed compression for cloud radio access networks. In: Information theory workshop (ITW), 2012 IEEE, IEEE, pp 242–246

  127. Park SH, Simeone O, Sahin O, Shamai S (2013) Robust and efficient distributed compression for cloud radio access networks. IEEE Trans Veh Technol 62(2):692–703

    Article  MATH  Google Scholar 

  128. Park SH, Simeone O, Sahin O, Shamai S (2014) Robust layered transmission and compression for distributed uplink reception in cloud radio access networks. IEEE Trans Veh Technol 63(1):204–216

    Article  Google Scholar 

  129. Peng C, Xie P, Chen C (2014) Adjustable robust optimal dispatch of power system with large-scale photovoltaic power stations. Proc CSEE 34(25):4324–4332

    Google Scholar 

  130. Peng C, Xie P, Pan L, Yu R (2016) Flexible robust optimization dispatch for hybrid wind/photovoltaic/hydro/thermal power system. IEEE Trans Smart Grid 7(2):751–762

    Google Scholar 

  131. Perelman L, Housh M, Ostfeld A (2013) Explicit demand uncertainty formulation for robust design of water distribution systems. In: World environmental and water resources congress 2013: showcasing the future, pp 684–695

  132. Pham GVA (2015) Robust optimization and data approximation in machine learning. PhD thesis, UC Berkeley

  133. Postek K, Den Hertog D, Kind J, Pustjens C (2016) Adjustable robust strategies for flood protection. Omega 82:142–54

    Article  Google Scholar 

  134. Postek K, Den Hertog D, Kind J, Pustjens C (2019) Adjustable robust strategies for flood protection. Omega 82:142–154

    Article  Google Scholar 

  135. Qin B, Xia Y, Wang S, Du X (2011) A novel bayesian classification for uncertain data. Knowl-Based Syst 24(8):1151–1158

    Article  Google Scholar 

  136. Ren J, Lee SD, Chen X, Kao B, Cheng R, Cheung D (2009) Naive Bayes classification of uncertain data. In: Data mining, 2009. ICDM’09. Ninth IEEE international conference on, IEEE, pp 944–949

  137. Rodrigues F, Agra A, Christiansen M, Hvattum LM, Requejo C (2019) Comparing techniques for modelling uncertainty in a maritime inventory routing problem. Eur J Oper Res

  138. Sands TM, Tayal D, Morris ME, Monteiro ST (2015) Robust stock value prediction using support vector machines with particle swarm optimization. In: 2015 IEEE congress on evolutionary computation (CEC), IEEE, pp 3327–3331

  139. Santofimia JFG (2008) Optimization-based methods for classification and regression problems with imprecise data. Universidad de Sevilla, Sevilla

    Google Scholar 

  140. dos Santos Diz GS, Hamacher S, Oliveira F (2018) A robust optimization model for the maritime inventory routing problem. Flex Serv Manuf J 20:1–27

    Google Scholar 

  141. Schölkopf B, Platt JC, Shawe-Taylor J, Smola AJ, Williamson RC (2001) Estimating the support of a high-dimensional distribution. Neural Comput 13(7):1443–1471

    Article  MATH  Google Scholar 

  142. Shaham U, Yamada Y, Negahban S (2015) Understanding adversarial training: increasing local stability of neural nets through robust optimization. arXiv:151105432 (arXiv preprint)

  143. Shaham U, Yamada Y, Negahban S (2018) Understanding adversarial training: increasing local stability of supervised models through robust optimization. Neurocomputing

  144. Shahmansoori A, Seco-Granados G, Wymeersch H (2015) Robust power allocation for OFDM wireless network localization. In: 2015 IEEE international conference on communication workshop (ICCW), IEEE, pp 718–723

  145. Shi Y, Zhang J, Letaief KB (2015) Robust group sparse beamforming for multicast green cloud-ran with imperfect CSI. IEEE Trans Signal Process 63(17):4647–4659

    Article  MathSciNet  MATH  Google Scholar 

  146. Shishebori D, Babadi AY (2015) Robust and reliable medical services network design under uncertain environment and system disruptions. Transp Res Part E Logist Transp Rev 77:268–288

    Article  Google Scholar 

  147. Shivaswamy PK, Bhattacharyya C, Smola AJ (2006) Second order cone programming approaches for handling missing and uncertain data. J Mach Learn Res 7((Jul)):1283–1314

    MathSciNet  MATH  Google Scholar 

  148. Simchi-Levi D, Wang H, Wei Y (2018) Increasing supply chain robustness through process flexibility and inventory. Prod Oper Manag 27(8):1476–1491

    Article  Google Scholar 

  149. Sniedovich M (2016) From statistical decision theory to robust optimization: a maximin perspective on robust decision-making. Robustness analysis in decision aiding, optimization, and analytics. Springer, Berlin, pp 59–87

    Chapter  Google Scholar 

  150. Street A, Moreira A, Arroyo JM (2014) Energy and reserve scheduling under a joint generation and transmission security criterion: an adjustable robust optimization approach. IEEE Trans Power Syst 29(1):3–14

    Article  Google Scholar 

  151. Subramanyam A (2018) Robust optimization of vehicle routing problems under uncertainty. PhD thesis

  152. Sun H, Gao Z, Zhao F (2014a) Dynamic network design problem under demand uncertainty: an adjustable robust optimization approach. Discret Dyn Nat Soc 20:2014

    MathSciNet  MATH  Google Scholar 

  153. Sun Y, Yuan Y, Wang G (2014b) Extreme learning machine for classification over uncertain data. Neurocomputing 128:500–506

    Article  Google Scholar 

  154. Tan J, Quek TQ, He Q (2017) Robust optimization for energy efficiency in multicast downlink c-ran. In: 2017 IEEE wireless communications and networking conference (WCNC), IEEE, pp 1–6

  155. Tang F, Zhang L, Huang J, Yang W (2009) An affinely adjustable robust optimization approach to emergency logistics distribution under uncertain demands. In: Industrial engineering and engineering management, 2009. IEEM 2009. IEEE international conference on, IEEE, pp 1738–1742

  156. Teng Y, Zhao W (2017) Robust group sparse beamforming for dense c-rans with probabilistic sinr constraints. In: 2017 IEEE wireless communications and networking conference (WCNC), IEEE, pp 1–6

  157. Thatte AA, Viassolo DE, Xie L (2012) Robust bidding strategy for wind power plants and energy storage in electricity markets. In: Power and energy society general meeting, 2012 IEEE, IEEE, pp 1–7

  158. Thatte AA, Xie L, Viassolo DE, Singh S (2013) Risk measure based robust bidding strategy for arbitrage using a wind farm and energy storage. IEEE Trans Smart Grid 4(4):2191–2199

    Article  Google Scholar 

  159. Thatte AA, Sun XA, Xie L (2014) Robust optimization based economic dispatch for managing system ramp requirement. In: 2014 47th Hawaii international conference on system sciences, IEEE, pp 2344–2352

  160. Torn A, Zilinskas A (1989) Global optimization. Springer, New York

    Book  MATH  Google Scholar 

  161. Trafalis TB, Alwazzi SA (2010) Support vector machine classification with noisy data: a second order cone programming approach. Int J Gen Syst 39(7):757–781

    Article  MathSciNet  MATH  Google Scholar 

  162. Trafalis TB, Gilbert RC (2006) Robust classification and regression using support vector machines. Eur J Oper Res 173(3):893–909

    Article  MathSciNet  MATH  Google Scholar 

  163. Tripathi S, Govindaraju RS (2008) Engaging uncertainty in hydrologic data sets using principal component analysis: Banpca algorithm. Water Resour Res 44:10

    Article  Google Scholar 

  164. Tseng FS, Gu JF (2015) Robust beamforming design in miso interference channels with RVQ limited feedback. IEEE Trans Veh Technol 64(2):580–592

    Article  Google Scholar 

  165. Utkin LV, Chekh AI (2015) A new robust model of one-class classification by interval-valued training data using the triangular kernel. Neural Netw 69:99–110

    Article  MATH  Google Scholar 

  166. Utkin LV, Coolen FP (2011) Interval-valued regression and classification models in the framework of machine learning. ISIPTA 11:371–380

    Google Scholar 

  167. Utkin LV, Zhuk YA (2017) An one-class classification support vector machine model by interval-valued training data. Knowl-Based Syst 120:43–56

    Article  Google Scholar 

  168. Utkin LV, Chekh AI, Zhuk YA (2016) Binary classification svm-based algorithms with interval-valued training data using triangular and epanechnikov kernels. Neural Netw 80:53–66

    Article  MATH  Google Scholar 

  169. Vapnik V (1963) Pattern recognition using generalized portrait method. Autom Remote Control 24:774–780

    Google Scholar 

  170. Vo XT (2015) Learning with sparsity and uncertainty by difference of convex functions optimization. PhD thesis, Université de Lorraine

  171. Vujanic R, Mariéthoz S, Goulart P, Morari M (2012) Robust integer optimization and scheduling problems for large electricity consumers. In: American control conference (ACC), 2012, IEEE, pp 3108–3113

  172. Vujanic R, Goulart P, Morari M (2016) Robust optimization of schedules affected by uncertain events. J Optim Theory Appl 171(3):1033–1054

    Article  MathSciNet  MATH  Google Scholar 

  173. Wan C, Markine V, Dollevoet R (2016) Robust optimisation of railway crossing geometry. Veh Syst Dyn 54(5):617–637

    Article  Google Scholar 

  174. Wang X, Pardalos PM (2014) A survey of support vector machines with uncertainties. Ann Data Sci 1(3–4):293–309

    Article  Google Scholar 

  175. Wang X, Fan N, Pardalos PM (2018) Robust chance-constrained support vector machines with second-order moment information. Ann Oper Res 263(1–2):45–68

    Article  MathSciNet  MATH  Google Scholar 

  176. Wang Z, Ng DWK, Wong VW, Schober R (2017) Robust beamforming design in c-ran with sigmoidal utility and capacity-limited backhaul. IEEE Trans Wirel Commun 16(9):5583–5598

    Article  Google Scholar 

  177. Weide O, Ryan D, Ehrgott M (2010) An iterative approach to robust and integrated aircraft routing and crew scheduling. Comput Oper Res 37(5):833–844

    Article  MATH  Google Scholar 

  178. Wenzhuo Y (2016) Optimization and learning under uncertainty-a unified robustness perspective. PhD thesis, National University of Singapore (Singapore)

  179. Xanthopoulos P, Pardalos PM, Trafalis TB (2012) Robust data mining. Springer, Berlin

    MATH  Google Scholar 

  180. Xiong P, Jirutitijaroen P (2012) An adjustable robust optimization approach for unit commitment under outage contingencies. In: Power and energy society general meeting, 2012 IEEE, IEEE, pp 1–8

  181. Xu H, Mannor S (2012) Robustness and generalization. Mach Learn 86(3):391–423

    Article  MathSciNet  MATH  Google Scholar 

  182. Xu H, Mannor S, Caramanis C (2008) Robustness, risk, and regularization in support vector machines. CoRR, vol arXiv:08033490

  183. Xu H, Caramanis C, Mannor S (2009a) Robust regression and lasso. Advances in neural information processing. Systems, Berlin, pp 1801–1808

    Google Scholar 

  184. Xu H, Caramanis C, Mannor S (2009b) Robustness and regularization of support vector machines. J Mach Learn Res 10((Jul)):1485–1510

    MathSciNet  MATH  Google Scholar 

  185. Xu L, Crammer K, Schuurmans D (2006) Robust support vector machine training via convex outlier ablation. AAAI 6:536–542

    Google Scholar 

  186. Xu Y, Zhao X (2016) Robust adaptive power control for cognitive radio networks. IET Signal Proc 10(1):19–27

    Article  Google Scholar 

  187. Xu Y, Yang Z, Pan X (2017) A novel twin support-vector machine with pinball loss. IEEE Trans Neural Netw Learn Syst 28(2):359–370

    Article  MathSciNet  Google Scholar 

  188. Yan C, Kung J (2016) Robust aircraft routing. Transp Sci 52(1):118–33

    Article  Google Scholar 

  189. Yan J, Li J, Zhao L, Chen R (2015) Robust joint transmit beamforming with qos guarantees in time-asynchronous das. IEEE Trans Veh Technol 64(4):1506–1518

    Article  Google Scholar 

  190. Yan S, Wang H, Tang X, Liu J, Huang TS (2008) Regression from uncertain labels and its applications to soft biometrics. IEEE Trans Inf Forensics Secur 3(4):698–708

    Article  Google Scholar 

  191. Yeager M, Gregory B, Key C, Todd M (2019) On using robust mahalanobis distance estimations for feature discrimination in a damage detection scenario. Struct Health Monit 18(1):245–253

    Article  Google Scholar 

  192. Yin X, Sinopoli B (2014) Adaptive robust optimization for coordinated capacity and load control in data centers. In: Decision and control (CDC), 2014 IEEE 53rd annual conference on, IEEE, pp 5674–5679

  193. Yuan W, Wang J, Qiu F, Chen C, Kang C, Zeng B (2016) Robust optimization-based resilient distribution network planning against natural disasters. IEEE Trans Smart Grid 7(6):2817–2826

    Article  Google Scholar 

  194. Yun S, Caramanis C (2012) System-level optimization in wireless networks: managing interference and uncertainty via robust optimization. IEEE/ACM Trans Netw 20(2):339–352

    Article  Google Scholar 

  195. Zhang D, Li X, Huang Y, Li S, Qian Q (2015) A robust optimization model for green regional logistics network design with uncertainty in future logistics demand. Adv Mech Eng 7(12):1687814015620518

    Google Scholar 

  196. Zhang Q, Liu HH (2019) Robust nonlinear control of close formation flight. arXiv:190407479 (arXiv preprint)

  197. Zhang Q, Morari MF, Grossmann IE, Sundaramoorthy A, Pinto JM (2016) An adjustable robust optimization approach to scheduling of continuous industrial processes providing interruptible load. Comput Chem Eng 86:106–119

    Article  Google Scholar 

  198. Zhao X, Bi X, Wang G, Zhang Z, Yang H (2016) Uncertain xml documents classification using extreme learning machine. Neurocomputing 174:375–382

    Article  Google Scholar 

  199. Zhen J, Den Hertog D, Sim M (2016) Adjustable robust optimization via Fourier–Motzkin elimination. Optimization Online

  200. Zhou M, Zhao X (2018) A robust energy efficiency power allocation algorithm in cognitive radio networks. China Commun 15(10):150–158

    Article  Google Scholar 

Download references

Acknowledgements

The authors appreciate comments and suggestions by anonymous reviewers that have resulted in substantially increasing the quality of the paper. The first author would like to acknowledge the IIT BHU research fellowship Grant.

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, IIT BHU, Varanasi, India

    Manisha Singla & K. K. Shukla

  2. Department of Mathematical Sciences, IIT BHU, Varanasi, India

    Debdas Ghosh

Authors
  1. Manisha Singla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Debdas Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. K. Shukla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manisha Singla.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singla, M., Ghosh, D. & Shukla, K.K. A survey of robust optimization based machine learning with special reference to support vector machines. Int. J. Mach. Learn. & Cyber. 11, 1359–1385 (2020). https://doi.org/10.1007/s13042-019-01044-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-019-01044-y

Keywords

Search

Navigation