Conformal Prediction (CP) is a powerful statistical machine learning tool to construct uncertainty sets with coverage guarantees, which has fueled its extensive adoption in generating prediction regions for decision-making tasks, e.g., Trajectory Optimization (TO) in uncertain environments. However, existing methods predominantly employ a sequential scheme, where decisions rely unidirectionally on the prediction regions, and consequently the information from decision-making fails to be fed back to instruct CP. In this paper, we propose a novel Feedback-Based CP (Fb-CP) framework for shrinking-horizon TO with a joint risk constraint over the entire mission time. Specifically, a CP-based posterior risk calculation method is developed by fully leveraging the realized trajectories to adjust the posterior allowable risk, which is then allocated to future times to update prediction regions. In this way, the information in the realized trajectories is continuously fed back to the CP, enabling attractive feedback-based adjustments of the prediction regions and a provable online improvement in trajectory performance. Furthermore, we theoretically prove that such adjustments consistently maintain the coverage guarantees of the prediction regions, thereby ensuring provable safety. Additionally, we develop a decision-focused iterative risk allocation algorithm with theoretical convergence analysis for allocating the posterior allowable risk which closely aligns with Fb-CP. Furthermore, we extend the proposed method to handle distribution shift. The effectiveness and superiority of the proposed method are demonstrated through benchmark experiments.

In recent years, there has been a growing research interest in decision-focused learning, which embeds optimization problems as a layer in learning pipelines and demonstrates a superior performance than the prediction-focused approach. However, for distributionally robust optimization (DRO), a popular paradigm for decision-making under uncertainty, it is still unknown how to embed it as a layer, i.e., how to differentiate decisions with respect to an ambiguity set. In this paper, we develop such differentiable DRO layers for generic mixed-integer DRO problems with parameterized second-order conic ambiguity sets and discuss its extension to Wasserstein ambiguity sets. To differentiate the mixed-integer decisions, we propose a novel dual-view methodology by handling continuous and discrete parts of decisions via different principles. Specifically, we construct a differentiable energy-based surrogate to implement the dual-view methodology and use importance sampling to estimate its gradient. We further prove that such a surrogate enjoys the asymptotic convergency under regularization. As an application of the proposed differentiable DRO layers, we develop a novel decision-focused learning pipeline for contextual distributionally robust decision-making tasks and compare it with the prediction-focused approach in experiments

Collision-free navigation is a critical issue in robotic systems as the environment is often dynamic and uncertain. This paper investigates a data-stream-driven motion control problem for mobile robots to avoid randomly moving obstacles when the probability distribution of the obstacle’s movement is partially observable through data and can be even time-varying. A data-stream-driven ambiguity set is firstly constructed from movement data by leveraging a Dirichlet process mixture model and is updated online using real-time data. Then we propose an Online-Learning-based Distributionally Robust Nonlinear Model Predictive Control (OL-DR-NMPC) approach for limiting the risk of collision through considering the worst-case distribution within the ambiguity set. To facilitate solving the OL-DR-NMPC problem, we reformulate it as a finite-dimensional nonlinear optimization problem. To cope with the bilinear matrix inequality constraints in the nonlinear problem, we develop a parabolic relaxation and a sequential algorithm, by which the problem is further transformed into polynomial-time solvable surrogates. The simulations using a quadrotor model are employed to demonstrate the effectiveness and advantages of the proposed method.

The hydrogen-based multimicrogrid (HMMG) has emerged as a game changer for energy transition. However, it encounters new challenges in tackling uncertainty data streams stemming from intermittent renewable energy and load. This article presents a multiple-time-scale HMMG energy management framework. In the day-ahead stage, the optimal scheduling is determined. The deviations of day-ahead predictions are redressed by intraday rescheduling. To accommodate the uncertainty data streams, a novel data-stream-driven distributionally robust model predictive control is proposed for the HMMG real-time operation. Specifically, a Dirichlet process mixture model is leveraged to construct an online-updated ambiguity set, which adequately characterizes the multimodality and local moment information of uncertainties. Based upon this ambiguity set, the data-stream-driven distributionally robust model predictive control enhances the real-time tracking performance of energy storage references. Its salient feature is the capability of greatly reducing conservatism while ensuring probabilistic operational constraints even under time-varying uncertainty distributions. Since this real-time operation is an intractable infinite-dimensional optimization problem, a novel constraint-tightening technique is proposed to address the computational challenge. Case studies demonstrate that the proposed approach offers advantages over state-of-the-art methods in out-of-sample performance.