Overview of Problem Formulations and Optimization Algorithms in the Presence of Uncertainty

Optimization under uncertainty is a key problem in order to solve complex system design problem while taking into account inherent physical stochastic phenomena, lack of knowledge, modeling simplifications, etc. Different reviews of optimization techniques in the presence of uncertainty can be found in the literature. The choice of the algorithm is often problem-dependent. The designer has to choose firstly the optimization problem formulation with respect to the system specifications and study but also the optimization algorithm to apply. The objective of this chapter is to present the different existing approaches to solve an optimization problem under uncertainty and to focus specifically on the uncertainty handling mechanisms. The chapter is organized as follows. Firstly, in Section 5.1, different optimization problem formulations are introduced, highlighting the importance of uncertainty measures and the distinctions between robustness-based formulation, reliability-based formulation, and robustness-and-reliability-based formulation. Then, in Section 5.2, different approaches to quantify the uncertainty in optimization are discussed. Finally, in the Section 5.3, an overview of optimization algorithms is presented with a focus on stochastic gradient, population-based algorithms, and surrogate-based approaches. For each type of algorithms the handling of uncertainty is analyzed and discussed.