Computational Optimization and Applications

, Volume 59, Issue 3, pp 617–638 | Cite as

A primal-dual aggregation algorithm for minimizing conditional value-at-risk in linear programs

  • Daniel Espinoza
  • Eduardo Moreno


Recent years have seen growing interest in coherent risk measures, especially in Conditional Value-at-Risk (\(\mathrm {CVaR}\)). Since \(\mathrm {CVaR}\) is a convex function, it is suitable as an objective for optimization problems when we desire to minimize risk. In the case that the underlying distribution has discrete support, this problem can be formulated as a linear programming (LP) problem. Over more general distributions, recent techniques, such as the sample average approximation method, allow to approximate the solution by solving a series of sampled problems, although the latter approach may require a large number of samples when the risk measures concentrate on the tail of the underlying distributions. In this paper we propose an automatic primal-dual aggregation scheme to exactly solve these special structured LPs with a very large number of scenarios. The algorithm aggregates scenarios and constraints in order to solve a smaller problem, which is automatically disaggregated using the information of its dual variables. We compare this algorithm with other common approaches found in related literature, such as an improved formulation of the full problem, cut-generation schemes and other problem-specific approaches available in commercial software. Extensive computational experiments are performed on portfolio and general LP instances.


Conditional value at risk Aggregation techniques  Approximation methods Sample average approximation 



Daniel Espinoza was partially funded by the FONDECYT Grant 1110024 and Millenium Nucleus Information and Coordination in Networks ICM/FIC P10-024F. Eduardo Moreno acknowledges the financial support of the FONDECYT Grant 1130681 and Basal Center CMM-UCh. Both authors acknowledge Stan Uryasev for providing AORDA software for benchmarking purposes.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Industrial EngineeringUniversidad de ChileSantiagoChile
  2. 2.Faculty of Engineering and ScienceUniversidad Adolfo IbañezSantiagoChile

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