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 a dynamic measure of risk. For this reason, we propose a scenario tree decomposition approach, namely group subproblem approach, to obtain bounds for such problems with an objective of dynamic mean-CVaR risk measure. 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 di erent convolution of mean-CVaR risk measures and di erent 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 an 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 di erent choices of lower bounds and partition strategies. Comparison of the proposed algorithm and the commercial solver revealed that, on the average, the proposed algorithm yields 2.58 times stronger bounds compared to a commercial solver.