Skip to main content
Log in

Uncontrolled inexact information within bundle methods

  • Original Paper
  • Published:
EURO Journal on Computational Optimization


We consider convex non-smooth optimization problems where additional information with uncontrolled accuracy is readily available. It is often the case when the objective function is itself the output of an optimization solver, as for large-scale energy optimization problems tackled by decomposition. In this paper, we study how to incorporate the uncontrolled linearizations into (proximal and level) bundle algorithms in view of generating better iterates and possibly accelerating the methods. We provide the convergence analysis of the algorithms using uncontrolled linearizations, and we present numerical illustrations showing they indeed speed up resolution of two stochastic optimization problems coming from energy optimization (two-stage linear problems and chance-constrained problems in reservoir management).

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

Access this article

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

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others




  • Brannlund U, Kiwiel KC, Lindberg PO (1995) A descent proximal level bundle method for convex nondifferentiable optimization. Oper Res Lett 17(3):121–126

    Article  Google Scholar 

  • Beltran C, Tadonki C, Vial JPh (2006) Solving the p-median problem with a semi-lagrangian relaxation. Comput Optim Appl 35(2):239–260

  • Deák I (2006) Two-stage stochastic problems with correlated normal variables: computational experiences. Ann OR 142(1):79–97

    Article  Google Scholar 

  • Desrosiers J, Lbbecke ME (2005) A primer in column generation. In: Desaulniers G, Desrosiers J, Solomon M (eds) Column generation. Springer, US, pp 1–32 (English)

  • Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213

    Article  Google Scholar 

  • Dentcheva D, Martinez G (2013) Regularization methods for optimization problems with probabilistic constraints. Math Program (Ser A) 138(1–2):223–251

    Article  Google Scholar 

  • de Oliveira W, Sagastizábal C (2014) Level bundle methods for oracles with on-demand accuracy. Optim Methods Softw 29(6):1180–1209

    Article  Google Scholar 

  • de Oliveira W, Sagastizábal C, Lemaréchal C (2014) Convex proximal bundle methods in depth: a unified analysis for inexact oracles. Math Program 148(1–2):241–277 (English)

    Article  Google Scholar 

  • de Oliveira W, Sagastizábal C, Penna D, Maceira M, Damázio J (2010) Optimal scenario tree reduction for stochastic streamflows in power generation planning problems. Optim Methods Softw 25(6):917–936

    Article  Google Scholar 

  • de Oliveira W, Sagastizábal C, Scheimberg S (2011) Inexact bundle methods for two-stage stochastic programming. SIAM J Optim 21(2):517–544

    Article  Google Scholar 

  • de Oliveira W, Solodov M (2015) A doubly stabilized bundle method for nonsmooth convex optimization. Math Program 1–35. doi:10.1007/s10107-015-0873-6

  • Fábián C (2000) Bundle-type methods for inexact data. Cent Eur J Oper Res 8:35–55

    Google Scholar 

  • Frangioni A (2002) Generalized bundle methods. SIAM J Optim 13(1):117–156

  • Geoffrion AM (1972) Generalized Benders decomposition. J Optim Theory Appl 10(4):237–260

    Article  Google Scholar 

  • Hintermüller M (2001) A proximal bundle method based on approximate subgradients. Comput Optim Appl 20:245–266. doi:10.1023/A:1011259017643

    Article  Google Scholar 

  • Hiriart-Urruty J-B, Lemaréchal C (1993) Convex analysis and minimization algorithms. Grund. der math. Wiss, Springer-Verlag, pp 305–306 (two volumes)

  • Kelley JE (1960) The cutting plane method for solving convex programs. J Soc Ind Appl Math 8:703–712

    Article  Google Scholar 

  • Kiwiel KC (2006) A proximal bundle method with approximate subgradient linearizations. SIAM J Optim 16(4):1007–1023

    Article  Google Scholar 

  • Lemaréchal C (2001) Lagrangian relaxation. In: Jünger M, Naddef D (eds) Computational combinatorial optimization. Springer Verlag, Heidelberg, pp 112–156

    Chapter  Google Scholar 

  • Lemaréchal C, Nemirovskii A, Nesterov Y (1995) New variants of bundle methods. Math Program 69(1):111–147

    Article  Google Scholar 

  • Magnanti TL, Wong RT (1981) Accelerating benders decomposition: algorithmic enhancement and model selection criteria. Oper Res 29(3):464–484

    Article  Google Scholar 

  • Ruszczyński A, Shapiro A (2003) Stochastic programming. Handbooks in operations research and management science, vol 10. Elsevier, Amsterdam

    Google Scholar 

  • Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory. MPS-SIAM series on optimization. SIAM—Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia

  • Solodov MV (2003) On approximations with finite precision in bundle methods for nonsmooth optimization. J Optim Theory Appl 119(1):151–165

    Article  Google Scholar 

  • Tahanan M, van Ackooij W, Frangioni A, Lacalandra F (2015) Large-scale unit commitment under uncertainty. 4OR-Q J Oper Res 13(2):115–171

  • van Ackooij W, Berge V, de Oliveira W, Sagastizábal C (2015) Probabilistic optimization via approximate p-efficient points and bundle methods. Tech. report. Optimization Online report number 4927

  • van Ackooij W, de Oliveira W (2014) Level bundle methods for constrained convex optimization with various oracles. Comput Optim Appl 57(3):555–597

    Article  Google Scholar 

  • van Ackooij W, Frangioni A, Oliveira W (2015) Inexact stabilized benders decomposition approaches to chance-constrained problems with finite support (Submitted). Available as preprint TR-15-01 of Universita di Pisa Dipartimento di Informatica

  • van Ackooij W, Henrion R, Moller A, Zorgati R (2014) Joint chance constrained programming for hydro reservoir management. Optim Eng 15(2):509–531

    Google Scholar 

  • Zakeri G, Philpott A, Ryan D (2000) Inexact cuts in benders decomposition. SIAM J Optim 10(3):643–657

    Article  Google Scholar 

Download references


We thank Antonio Frangioni (Univ. of Pisa, Italy) for insightful discussions on a first version of this article and Wim van Ackooij (EDF, France) for providing us with the real-life data set used in Sect. 4.2. The first author gratefully acknowledges the support of the Grant “ANR GeoLMI” and the CNRS Mastodons project “gargantua/titan”. The second author gratefully acknowledges the support provided by Severo Ochoa Program SEV-2013-0323 and Basque Government BERC Program 2014-2017.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jérôme Malick.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Malick, J., de Oliveira, W. & Zaourar, S. Uncontrolled inexact information within bundle methods. EURO J Comput Optim 5, 5–29 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification