New a priori and a posteriori probabilistic bounds for robust counterpart optimization: III. Exact and near-exact a posteriori expressions for known probability distributions

The performance of robust optimization is closely connected with probabilistic bounds that determine the probability of constraint violation due to uncertain parameter realizations. In Part I of this work, new a priori and a posteriori probabilistic bounds were developed for cases when robust optimization is applied to uncertain optimization problems with parameters whose probability distributions were unknown. In Part II, the focus shifted to known probability distributions and a priori bounds. In this paper, new, tight a posteriori expressions are developed for constraints containing parameters with specific known distributions, that is, those attributed normal, uniform, discrete, gamma, chi-squared, Erlang, or exponential distributions. The nature of some of the expressions requires efficient implementations, and new algorithmic methods are discussed which greatly improve applicability. These new expressions are much tighter than existing bounds and greatly reduce the conservatism of robust solutions. The theoretical and algorithmic results of Parts I, II, and III allow for wider usage of robust optimization in process synthesis and operations research applications.