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Distributionally robust control and optimization


B. P. Van Parys


The principal aim of this dissertation is to discuss and advance the use of distributionally robust constraints for decision-making in an uncertain environment. Distributionally robust constraints are the robust counterpart of uncertain constraints subject to a random outcome of which the distribution is only partially known. This dissertation will study in particular control and optimization problems for which these type of constraints constitute sensible design objectives. The contributions of this thesis fall into three categories each of which support the principal aim by taking down a hurdle which prevents its attainment. We briefly discuss the aforementioned categories and the contributions of this dissertation therein.

First, we argue that distributionally robust constraints present sound design objectives which are often more practically relevant than the classical worst-case or chance constrained alternatives. Distributionally robust constraints are often less pessimistic and do not require the distribution of the disturbances involved to be known exactly. In practice distributions are indeed never observed directly, but rather need to be estimated from noisy historical data. We consider two types of distributionally robust constraints in this dissertation. In the first type, we require that constraints hold with a given probability for all disturbance distributions consistent with the known partial distributional information. These constraints are referred to as distributionally robust chance constraints. In a second type of constraints, referred to as distributionally robust CVaR constraints, we additionally require the expected constraint violation to be small for all relevant disturbance distributions. Either constraint type is discussed and promoted as sound and sensible design objectives for both static optimization and dynamic control problems.

Second, in many interesting situations distributionally robust constraints are amendable to practical computation. The need for computational tools applicable to distributionally robust constraints naturally leads to the study of uncertainty quantification problems in which a probabilistic question needs to be answered using only limited statistical information. Uncertainty quantification problems find their roots in the classical univariate probability inequalities advanced by the Russian school of probability (Chebyshev, Markov, Lyapunov and Bernstein) and their origins can be traced back to the middle of the 19th century. In this dissertation these classical Chebyshev type inequalities are generalized to bounds on the probability of events in arbitrary dimensions based solely on second-order moment information. Instead of a closed form solution, these bounds are stated in terms of a tractable convex optimization problem. We discuss why Chebyshev type bound are achieved by pathological discrete distributions which render the corresponding inequalities overly pessimistic. In an attempt to exclude these practically irrelevant distributions, Gauss type probability inequalities and uncertainty quantification problems will be at the center of attention. In aforementioned Gauss bounds, the considered distributions are required to enjoy further structural properties which many practical distributions possess such as unimodality or monotonicity.

We indicate lastly that all discussed problems can be treated in a unified fashion and stated in the language of convex optimization. This dissertation indeed brings together and merges many relevant results in probability theory by unveiling their innate convex nature. By presenting a deep analogy between vectors in $\Re^n$ and probability distributions on $\Re^n$, it will be argued that the same mathematical tools used in the analysis of classical worst-case robust constraints can be wielded in our distributionally robust setting equally well. Many results found in this dissertation concerning probability theory and uncertainty quantification problems have indeed a direct counterpart in either convex analysis or optimization, respectively.



Type of Publication:

(03)Ph.D. Thesis

M. Morari

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% Autogenerated BibTeX entry
@PhDThesis { Xxx:2015:IFA_5354,
    author={B. P. Van Parys},
    title={{Distributionally robust control and optimization}},
    school={ETH Zurich},
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