From Uncertainty Data to Robust Policies for Temporal Logic Planning

We consider the problem of synthesizing robust disturbance feedback policies for systems performing complex tasks. We formulate the tasks as linear temporal logic specifications and encode them into an optimization framework via mixed-integer constraints. Both the system dynamics and the specifications are known but affected by uncertainty. The distribution of the uncertainty is unknown, however realizations can be obtained. We introduce a data-driven approach where the constraints are fulfilled for a set of realizations and provide probabilistic generalization guarantees as a function of the number of considered realizations. We use separate chance constraints for the satisfaction of the specification and operational constraints. This allows us to quantify their violation probabilities independently. We compute disturbance feedback policies as solutions of mixed-integer linear or quadratic optimization problems. By using feedback we can exploit information of past realizations and provide feasibility for a wider range of situations compared to static input sequences. We demonstrate the proposed method on two robust motion-planning case studies for autonomous driving

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Robust optimization with incremental recourse

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem

A scenario approach for non-convex control design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in terms of performance and require high sample complexity to achieve the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control- design problems that can be addressed via randomization, we apply our scenario approach to randomized Model Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro