Nonlinear chance-constrained problems with applications to hydro scheduling

  • Andrea LodiEmail author
  • Enrico Malaguti
  • Giacomo Nannicini
  • Dimitri Thomopulos
Full Length Paper Series B


We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.


Chance-constraints Outer approximation Benders decomposition Branch-and-Cut Hydro scheduling 

Mathematics Subject Classification

90C15 Stochastic Programming 49M27 Decomposition Methods 90C57 Branch-and-Cut 



The authors are extremely grateful to Costas Baslis and Anastasios Bakirtzis for sharing the data on the Greek power system discussed in [22], to Alberto Borghetti for several helpful discussions, to the anonymous referees and the associate editor for their comments that helped significantly improve the paper. The first and fourth authors acknowledge the support of MIUR, Italy, under the Grant PRIN 2012. The second author acknowledges the support of the Air Force Office of Scientific Research under Award Number FA9550- 17-1-0025. Traveling support by the EU ITN 316647 “Mixed-Integer Nonlinear Optimization” is acknowledged by the third author. Part of this research was carried out at the Singapore University of Technology and Design, supported by grant SRES11012 and IDC grant IDSF1200108.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Canada Excellence Research Chair, École Polytechnique de MontréalMontrealCanada
  2. 2.DEI, Università di BolognaBolognaItaly
  3. 3.IBM T.J. WatsonYorktown HeightsUSA
  4. 4.DESTEC, Università di PisaPisaItaly

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