Bounds on risk-averse mixed-integer multi-stage stochastic programming problems with mean-CVaR

Risk-averse mixed-integer multi-stage stochastic programming forms a class of extremely challenging problems since the problem size grows exponentially with the number of stages, the problem is non convex due to integrality restrictions, and the objective function is nonlinear in general. We propose a scenario tree decomposition approach, namely group subproblem approach, to obtain bounds for such problems with an objective of dynamic mean conditional value-at-risk (mean-CVaR). Our approach does not require any special problem structure such as convexity and linearity, therefore it can be applied to a wide range of problems. We obtain lower bounds by using different convolution of mean-CVaR risk measures and different scenario partition strategies. The upper bounds are obtained through the use of optimal solutions of group subproblems. Using these lower and upper bounds, we propose a solution algorithm for risk-averse mixed-integer multi-stage stochastic problems with mean-CVaR risk measures. We test the performance of the proposed algorithm on a multi-stage stochastic lot sizing problem and compare different choices of lower bounds and partition strategies. Comparison of the proposed algorithm to a commercial solver revealed that, on the average, the proposed algorithm yields 1.13% stronger bounds. The commercial solver requires additional running time more than a factor of five, on the average, to reach the same optimality gap obtained by the proposed algorithm. (C) 2017 Elsevier B.V. All rights reserved.