A Multicut Approach to Compute Upper Bounds for Risk-Averse SDDP

TL;DR

This paper introduces a simple, efficient multicut SDDP-based method to compute upper bounds for risk-averse stochastic optimization, improving convergence and scenario exploration with minimal algorithm modifications.

Contribution

It proposes leveraging multicut SDDP to estimate upper bounds for CVaR-based models, enhancing efficiency and accuracy without significant computational overhead.

Findings

Method accurately computes objective values in test cases.

Upper bounds are consistently higher than lower bounds.

Scenario exploration is improved, leading to better bounds.

Abstract

Stochastic Dual Dynamic Programming (SDDP) is a widely used and fundamental algorithm for solving multistage stochastic optimization problems. Although SDDP has been frequently applied to solve risk-averse models with the Conditional Value-at-Risk (CVaR), it is known that the estimation of upper bounds is a methodological challenge, and many methods are computationally intensive. In practice, this leaves most SDDP implementations without a practical and clear stopping criterion. In this paper, we propose using the information already contained in a multicut formulation of SDDP to solve this problem with a simple and computationally efficient methodology. The multicut version of SDDP, in contrast with the typical average cut, preserves the information about which scenarios give rise to the worst costs, thus contributing to the CVaR value. We use this fact to modify the standard sampling method on the forward step so the average of multiple paths approximates the nested CVaR cost. We highlight that minimal changes are required in the SDDP algorithm and there is no additional computational burden for a fixed number of iterations. We present multiple case studies to empirically demonstrate the effectiveness of the method. First, we use a small hydrothermal dispatch test case, in which we can write the deterministic equivalent of the entire scenario tree to show that the method perfectly computes the correct objective values. Then, we present results using a standard approximation of the Brazilian operation problem and a real hydrothermal dispatch case based on data from Colombia. Our numerical experiments showed that this method consistently calculates upper bounds higher than lower bounds for those risk-averse problems and that lower bounds are improved thanks to the better exploration of the scenarios tree.