New Solution Methods for Joint Chance-Constrained Stochastic Programs with Random Left-Hand Sides

We consider joint chance-constrained programs with random lefthand sides. The motivation of this project is that this class of problem has many important applications, but there are few existing solution methods. For the most part, we deal with the subclass of problems for which the underlying parameter distributions are discrete. This assumption allows the original problem to be formulated as a deterministic equivalent mixed-integer program. We rst approach the problem as a mixed-integer program and derive a class of optimality cuts based on irreducibly infeasible subsets of the constraints of the scenarios of the problem. The IIS cuts can be computed effciently by means of a linear program. We give a method for improving the upper bound of the problem when no IIS cut can be identifi ed. We also give an implementation of an algorithm incorporating these ideas and finish with some computational results. We present a tabu search metaheuristic for fi nding good feasible solutions to the mixed-integer formulation of the problem. Our heuristic works by de ning a sufficient set of scenarios with the characteristic that all other scenarios do not have to be considered when generating upper bounds. We then use tabu search on the one-opt neighborhood of the problem. We give computational results that show our metaheuristic outperforming the state-of-the-art industrial solvers. We then show how to reformulate the problem so that the chance-constraints are monotonic functions. We then derive a convergent global branch-and-bound algorithm using the principles of monotonic optimization. We give a finitely convergent modi cation of the algorithm. Finally, we give a discussion on why this algorithm is computationally ine ffective. The last section of this dissertation details an application of joint chance-constrained stochastic programs to a vaccination allocation problem. We show why it is necessary to formulate the problem with random parameters and also why chance-constraints are a good framework for de fining an optimal policy. We give an example of the problem formulated as a chance constraint and a short numerical example to illustrate the concepts

Optimistic and pessimistic ambiguous chance constraints with applications

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.Open Acces