We demonstrate how adjustable robust optimization (ARO) problems with fixed recourse can be casted as static robust optimization problems via Fourier-Motzkin elimination (FME). Through the lens of FME, we characterize the structures of the optimal decision rules for a broad class of ARO problems. A scheme based on a blending of classical FME and a simple Linear Programming technique that can efficiently remove redundant constraints, is developed to reformulate ARO problems. This generic reformulation technique enhances the classical approximation scheme via decision rules, and enables us to solve adjustable optimization problems to optimality. We show via numerical experiments that, for small-size ARO problems our novel approach finds the optimal solution. For moderate or large-size instances, we eliminate a subset of the adjustable variables, which improves the solutions from decision rule approximations.

We introduce a general framework for robust dataenabled predictive control (DeePC) for linear time-invariant (LTI) systems. The proposed framework enables us to obtain model-free optimal control for LTI systems based on noisy input/output data. More specifically, robust DeePC solves a min-max optimization problem to compute the optimal control sequence that is resilient to all possible realizations of the uncertainties in the input/output data within a prescribed uncertainty set. We present computationally tractable reformulations of the min-max problem with various uncertainty sets. Furthermore, we show that even though an accurate prediction of the future behavior is unattainable in practice due to inaccessibility of the perfect input/output data, the obtained robust optimal control sequence provides performance guarantees for the actually realized input/output cost. We further show that the robust DeePC generalizes and robustifies the regularized DeePC (with quadratic regularization or 1-norm regularization) proposed in the literature. Finally, we demonstrate the performance of the proposed robust DeePC algorithm on high-fidelity, nonlinear, and noisy simulations of a grid-connected power converter system.

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We introduce a novel scheme based on a blending of Fourier-Motzkin elimination (FME) and adjustable robust optimization techniques to compute the maximum volume inscribed ellipsoid (MVE) in a polytopic projection. It is well-known that deriving an explicit description of a projected polytope is NP-hard. Our approach does not require an explicit description of the projection, and can easily be generalized to find a maximally sized convex body of a polytopic projection. Our obtained MVE is an inner approximation of the projected polytope, and its center is a centralized relative interior point of the projection. Since FME may produce many redundant constraints, we apply an LP-based procedure to keep the description of the projected polytopes at its minimal size. Furthermore, we propose an upper bounding scheme to evaluate the quality of the inner approximations. We test our approach on a simple polytope and a color tube design problem, and observe that as more auxiliary variables are eliminated, our inner approximations and upper bounds converge to optimal solutions.

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