We introduce a verification framework to numerically analyze inexact model predictive controllers (MPCs) in the constrained non-linear discrete-time setting. Rather than modifying the controller so that guarantees hold by construction, we treat the controller as given. In particular, we focus on two types of inexact controllers: (a) one whose input is extracted from a primal-dual point satisfying the Karush-Kuhn-Tucker (KKT) conditions of the non-convex MPC problem, and (b) one whose input is obtained by linearizing the dynamics and solving a convex quadratic program. The main idea of our verification framework is to formulate an optimization problem that searches over the worst-case initial state within a given set and control inputs consistent with the inexact controller to maximize a carefully-chosen performance metric. Using this framework, we show how to certify (i) the worst-case suboptimality gap of a single MPC problem, (ii) the worst-case closed-loop suboptimality gap over a given number of dynamical system iterations, (iii) closed-loop stability, and (iv) feasibility of the closed-loop system. Through numerical examples, we showcase the ability of our framework to precisely quantify both types of suboptimality, and to test the stability and feasibility of the inexact controllers.
Rajiv Sambharya, S. C. Anand, George J. Pappas· 0 citations
Abstract.
We introduce a machine-learning framework to learn the hyperparameter sequence of first-order methods (e.g. , the step sizes in gradient descent) to quickly solve parametric convex optimization problems. Our computational architecture amounts to running fixed-point iterations where the hyperparameters are the same across all parametric instances and consists of two phases. In the first step-varying phase the hyperparameters vary across iterations, while in the second steady-state phase the hyperparameters are constant across iterations. Our learned optimizer is flexible, in that it can be evaluated on any number of iterations and is guaranteed to converge to an optimal solution. To train, we minimize the mean square error to a ground truth solution. In the case of gradient descent, the one-step optimal step size is the solution to a least squares problem, and in the case of unconstrained quadratic minimization, we can compute the two-step and three-step optimal solutions in closed form. In other cases, we backpropagate through the algorithm steps to minimize the training objective after a given number of steps. We show how to learn hyperparameters for several popular algorithms: gradient descent, proximal gradient descent, and two ADMM-based solvers: OSQP and SCS. We use a sample convergence bound to obtain generalization guarantees for the performance of our learned algorithm for unseen data, providing both lower and upper bounds. We showcase the effectiveness of our method with many examples, including ones from control, signal processing, and machine learning. Remarkably, our approach is highly data-efficient in that we only use 10 problem instances to train the hyperparameters in all of our examples.
Rajiv Sambharya, Bartolomeo Stellato· SIAM Journal on Mathematics...· 0 citations
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