Scenario tree reduction algorithms combining quantitative stability with k-medoids clustering for large-scale risk-averse multi-stage stochastics programming problems

Abstract

In this paper, we propose a novel scenario tree reduction framework derived from the quantitative stability of risk-averse multi-stage stochastic programs. We first establish the quantitative stability theorem for risk-averse multi-stage stochastic programming problems, a general scenario tree reduction model is then developed from the obtained quantitative bound, and finally we derive an error bound on the deviation between the optimal value of the multi-stage stochastic programming problem under the reduced scenario tree and that under the original scenario tree. Additionally, we introduce a new distance to measure the similarity among different nodes, and develop two scenario tree reduction algorithms utilizing the $k$-medoid and local search frameworks, respectively. The proposed scenario tree reduction algorithms can effectively control the error in the optimal value/decision of the optimization problem caused by the scenario tree reduction. Moreover, the low computational cost of these algorithms allows one to solve large-scale risk-averse multi-stage stochastic programs with many stages. Finally, a series of numerical experiments are carried out to demonstrate the superiority of proposed scenario tree reduction algorithms.