LC
L. Chaouach
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Decision-making under uncertainty is a fundamental challenge across many areas, such as engineering, finance, and healthcare. While stochastic optimization provides a principled framework by modeling uncertainty through probability distributions, identifying the correct distribution is not always straightforward; hence, a deeper layer of "uncertain" uncertainty emerges. This thesis addresses this challenge through distributionally robust optimization (DRO), which hedges decisions against an ambiguity set of plausible distributions consistent with observed data. The central contribution is the exploitation of structural prior knowledge, specifically the independence of uncertainty components , to construct tighter ambiguity sets. This reduces conservativeness while preserving rigorous statistical guarantees. Three interconnected contributions are developed in this work: (i) structured ambiguity sets tailored to independent uncertainty components; (ii) tractable reformulations and complexity-reduction procedures for the associated DRO problems; and (iii) a distributionally robust model predictive control scheme for linear systems under unknown disturbance distributions, in which the computational burden of DRO is handled entirely offline, yielding a practically implementable controller. The proposed framework enables reliable, high-performance data-driven decisions in settings where the true probability distribution is unknown but partial structural information about the independence of the uncertainty components is available.
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Decision-making under uncertainty is a fundamental challenge across many areas, such as engineering, finance, and healthcare. While stochastic optimization provides a principled framework by modeling uncertainty through probability distributions, identifying the correct distribution is not always straightforward; hence, a deeper layer of "uncertain" uncertainty emerges. This thesis addresses this challenge through distributionally robust optimization (DRO), which hedges decisions against an ambiguity set of plausible distributions consistent with observed data. The central contribution is the exploitation of structural prior knowledge, specifically the independence of uncertainty components , to construct tighter ambiguity sets. This reduces conservativeness while preserving rigorous statistical guarantees. Three interconnected contributions are developed in this work: (i) structured ambiguity sets tailored to independent uncertainty components; (ii) tractable reformulations and complexity-reduction procedures for the associated DRO problems; and (iii) a distributionally robust model predictive control scheme for linear systems under unknown disturbance distributions, in which the computational burden of DRO is handled entirely offline, yielding a practically implementable controller. The proposed framework enables reliable, high-performance data-driven decisions in settings where the true probability distribution is unknown but partial structural information about the independence of the uncertainty components is available.
The aim of this paper is to compare two classes of structured ambiguity sets, which are data-driven and can reduce the conservativeness of their associated optimization problems. These two classes of structured sets, coined Wasserstein hyperrectangles and multi-transport hyperrectangles, are explored in their trade-offs in terms of reducing conservativeness and providing tractable reformulations. It follows that multi-transport hyperrectangles lead to tractable optimization problems for a significantly broader range of objective functions under a decent compromise in terms of conservativeness reduction. The results are illustrated in an uncertainty quantification case study.
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The aim of this paper is to compare two classes of structured ambiguity sets, which are data-driven and can reduce the conservativeness of their associated optimization problems. These two classes of structured sets, coined Wasserstein hyperrectangles and multi-transport hyperrectangles, are explored in their trade-offs in terms of reducing conservativeness and providing tractable reformulations. It follows that multi-transport hyperrectangles lead to tractable optimization problems for a significantly broader range of objective functions under a decent compromise in terms of conservativeness reduction. The results are illustrated in an uncertainty quantification case study.
Ambiguity sets of probability distributions are a prominent tool to hedge against distributional uncertainty in stochastic optimization. The aim of this paper is to build tight Wasserstein ambiguity sets for data-driven optimization problems. The method exploits independence between the distribution components to introduce structure in the ambiguity sets and speed up their shrinkage with the number of collected samples. Tractable reformulations of the stochastic optimization problems are derived for costs that are expressed as sums or products of functions that depend only on the individual distribution components. The statistical benefits of the approach are theoretically analyzed for compactly supported distributions and demonstrated in a numerical example.
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Ambiguity sets of probability distributions are a prominent tool to hedge against distributional uncertainty in stochastic optimization. The aim of this paper is to build tight Wasserstein ambiguity sets for data-driven optimization problems. The method exploits independence between the distribution components to introduce structure in the ambiguity sets and speed up their shrinkage with the number of collected samples. Tractable reformulations of the stochastic optimization problems are derived for costs that are expressed as sums or products of functions that depend only on the individual distribution components. The statistical benefits of the approach are theoretically analyzed for compactly supported distributions and demonstrated in a numerical example.
In this paper, the problem of stability, recursive feasibility and convergence conditions of stochastic model predictive control for linear discrete-time systems affected by a large class of correlated disturbances is addressed. A stochastic model predictive control that guarantees convergence, average cost bound and chance constraint satisfaction is developed. The results rely on the computation of probabilistic reachable and invariant sets using the notion of correlation bound. This control algorithm results from a tractable deterministic optimal control problem with a cost function that upper-bounds the expected quadratic cost of the predicted state trajectory and control sequence. The proposed methodology only relies on the assumption of the existence of bounds on the mean and the covariance matrices of the disturbance sequence.
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In this paper, the problem of stability, recursive feasibility and convergence conditions of stochastic model predictive control for linear discrete-time systems affected by a large class of correlated disturbances is addressed. A stochastic model predictive control that guarantees convergence, average cost bound and chance constraint satisfaction is developed. The results rely on the computation of probabilistic reachable and invariant sets using the notion of correlation bound. This control algorithm results from a tractable deterministic optimal control problem with a cost function that upper-bounds the expected quadratic cost of the predicted state trajectory and control sequence. The proposed methodology only relies on the assumption of the existence of bounds on the mean and the covariance matrices of the disturbance sequence.