In this thesis, we consider optimisation problems which involve ambiguous chance constraints, i.e., probabilistic constraints where the probability distribution of the primitive uncertainties is at least partly unknown. In this case, we can define an ambiguity set that contains all distributions consistent with our prior knowledge of the uncertainty and take either a pessimistic (worst-case) or optimistic (best-case) view of the world. The former view can be used to actively optimise a system whilst guaranteeing some predefined level of safety; being robust even if the worst-case scenario materialises. The latter view can be used to actively optimise a system where it is required to reconstruct realisations of a random variable whose distribution is not known precisely.

We characterise the ambiguity set through generalised moment bounds and structural properties such as symmetry, unimodality, or independence patterns. Sufficient conditions are presented under which the corresponding chance constraints admit equivalent explicit tractable conic reformulations that can be solved with off-the-shelf solvers. However, in general, ambiguous chance constrained problems are provably difficult and we suggest efficiently computable conservative approximations.

To illustrate the effectiveness of our reformulations, we give two detailed and novel examples. First, we consider the pricing problem of a provider of cloud computing services. This provider faces uncertain demand and wishes to maximise profit, whilst maintaining a desired level of quality of service. We show that such a problem naturally fits within the pessimistic ambiguous chance constraint framework. Second, we consider the problem of improving the quality of a photographic image by reconstructing and then removing noise. We show that such a problem can be formulated as an optimistic ambiguous chance constrained program that generalises, and offers new insight to, an existing powerful image denoising approach.