The efficient computation of parametric solution sensitivities is a key challenge in the integration of learning-enhanced methods with nonlinear model predictive control, as their availability is crucial for many learning algorithms. This paper discusses the computation of solution sensitivities of general nonlinear programs using the implicit function theorem and smoothed optimality conditions treated in interior-point methods (IPM). We introduce sequential barrier quadratic programming (SBQP), which generalizes SQP with an IPM quadratic programming (QP) solver by allowing the barrier parameter to approach a prescribed nonnegative target value, recovering classical SQP when this target is zero. Previous works are limited to convex or unconstrained formulations, or lack an implementation for efficient sensitivity evaluation. The publication is accompanied by an efficient open-source implementation within the `acados` framework, providing both forward and adjoint sensitivities for general optimal control problems, achieving speedups exceeding 3x over the state-of-the-art solvers `mpc.pytorch` and `cvxpygen`.
We propose a robust Q-learning algorithm for Markov decision processes under model uncertainty when each state--action pair is associated with a finite ambiguity set of candidate transition kernels. This finite-measure framework enables user-designed uncertainty models, including competing fitted models and stress scenarios. {The learner observes successor states and their deterministic rewards from every candidate simulator, without requiring explicit knowledge of the transition probabilities or the reward function.} We establish almost sure convergence of the learned Q-function to the robust optimum, and derive a time-uniform high-probability error certificate that separates an observable empirical Bellman residual from transition-kernel estimation error. We also give quantitative finite-ambiguity approximation results for Wasserstein and parametric uncertainty models on finite state spaces. Numerical examples illustrate the effect of scenario selection and compare the tabular algorithm with robust value iteration.
High-quality matches between patients and primary care providers improve downstream health outcomes, yet provider turnover breaks these matches and forces patients to be reassigned. Health systems facing this problem must decide not only which providers to recommend, but how much choice to offer. Direct assignment avoids competition for scarce provider capacity but relies on noisy estimates of patient preferences; broader choice allows patients to correct these estimates using private information but can create costly coordination failures. We study this tradeoff through the design of fixed, personalized menus from which patients choose providers. We show that constructing the optimal menu is NP-hard and establish an information-theoretic lower bound. We find that offering each patient a single provider matches this bound up to a constant factor, while unrestricted choice can perform arbitrarily poorly. However, we show that additional choice improves on single-provider assignment when the system cannot reliably distinguish among a patient's alternatives. We formalize this tradeoff through two forms of loss: menu loss, which arises when menus offer too few options, and coordination loss, which arises when scarce capacity is exposed to too much choice. Motivated by this decomposition, we develop the Scenario-Averaged Marginals (SAM) algorithm, which tailors menus based on provider shadow prices so choice is offered only when utility exceeds the capacity cost. We identify sufficient conditions which SAM's regret matches the lower bound, namely when patients have valuable alternatives and offered menus don't create competition for scarce capacity. In a semi-synthetic environment calibrated to a provider network using Medicare data, SAM improves total utility by up to 19% over single-provider assignment as preference uncertainty grows, while improving utility for the least-served ZIP codes.
Robust subspace estimation is fundamental to many machine learning and data analysis tasks. Iteratively Reweighted Least Squares (IRLS) is an elegant and empirically effective approach to this problem, yet its theoretical properties remain poorly understood. This paper establishes that, under deterministic conditions, a variant of IRLS with dynamic smoothing regularization converges linearly to the underlying subspace from any initialization. We extend these guarantees to affine subspace estimation, a setting that lacks prior recovery theory. Additionally, we illustrate the practical benefits of IRLS through an application to low-dimensional neural network training. Our results provide the first global convergence guarantees for IRLS in robust subspace recovery and, more broadly, for nonconvex IRLS on a Riemannian manifold.