Robust Approximations to Joint Chance-constrained Problems

Two new approximate formulations to joint chance-constrained optimization problems are proposed in this paper. The relationships of CVaR (conditional-value-at-risk), chance constrains and robust optimization are reviewed. Firstly, two new upper bounds on E((·) ⁺) are proposed, where E stands for the expectation and x⁺ = max(0, x), based on which two approximate formulations for individual chance-constrained problems are derived. The approximations are proved to be the robust optimization with the corresponding uncertain sets. Then the approximations are extrapolated to joint chance-constrained problem. Finally numerical studies are performed to compare the solutions of individual and joint chance constraints approximations and the results demonstrate the validity of our method.