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

<Para ID="Par1">Recent years have seen growing interest in coherent risk measures, especially in Conditional Value-at-Risk (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathrm {CVaR}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="normal">CVaR</mi> </math> </EquationSource> </InlineEquation>). Since <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathrm {CVaR}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="normal">CVaR</mi> </math> </EquationSource> </InlineEquation> 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. Copyright Springer Science+Business Media New York 2014