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Alberto Bemporad

Publications and source records attributed to Alberto Bemporad.

At least 19 recordsLinked to original sources
arXiv2026

A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints

We propose an inexact proximal augmented Lagrangian method (P-ALM) for nonconvex structured optimization problems. The proposed method features an easily implementable rule not only for updating the penalty parameters, but also for adaptively tuning the proximal term. It allows the penalty parameter to grow rapidly in the early stages to speed up progress, while ameliorating the issue of ill-conditioning in later iterations, a well-known drawback of the traditional approach of linearly increasing the penalty parameters. A key element in our analysis lies in the observation that the augmented Lagrangian can be controlled effectively along the iterates, provided an initial feasible point is available. Our analysis, while simple, provides a new theoretical perspective about P-ALM and, as a by-product, results in similar convergence properties for its non-proximal variant, the classical augmented Lagrangian method (ALM). Numerical experiments, including convex and nonconvex problem instances, demonstrate the effectiveness of our approach.

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arXiv2026

Learning generalized Nash equilibria from pairwise preferences

Generalized Nash Equilibrium Problems (GNEPs) arise in many applications, including non-cooperative multi-agent control problems. Although many methods exist for finding generalized Nash equilibria, most of them rely on assuming knowledge of the objective functions or being able to query the best responses of the agents. We present a method for learning solutions of GNEPs only based on querying agents for their preference between two alternative decisions. We use the collected preference data to learn a GNEP whose equilibrium approximates a GNE of the underlying (unknown) problem. Preference queries are selected using an active-learning strategy that balances exploration of the decision space and exploitation of the learned GNEP. We present numerical results on game-theoretic linear quadratic regulation problems, as well as on other literature GNEP examples, showing the effectiveness of the proposed method.

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arXiv2026

Learning Parametric Monotone Games

We study the problem of learning from data a parametric Nash equilibrium (NE) problem that is monotone (or strongly monotone) for all parameter values. In the presence of local and shared convex constraints, monotonicity enables efficient computation of generalized Nash equilibria of the learned game. We consider two learning scenarios: (i) direct learning of the NE problem from samples of the agents' costs, and (ii) inverse learning of surrogate agents' costs from samples of their best responses. We propose two methods to solve these tasks. The first is a penalty-based approach that promotes monotonicity of the learned game during training. The second, based on a representation theorem we introduce for a broad class of monotone games, parameterizes the agents' costs so that monotonicity of the NE problem is guaranteed by construction, for all parameter values, regardless of the training data used. We illustrate the applicability of the proposed methods on several numerical examples. A Python library and the examples reported in the paper are available at https://github.com/bemporad/learn_monotone_games.

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arXiv2026

Solving Monotone Linear-Quadratic Generalized Nash Equilibrium Problems via Quadratic Programming

We consider generalized Nash equilibrium problems among $N$ players with convex quadratic costs and shared affine constraints, assuming only that the game's pseudogradient is merely monotone. We show that computing a variational generalized Nash equilibrium (v-GNE) is equivalent to solving a single convex quadratic program (QP) derived from the players' joint Karush--Kuhn--Tucker conditions. Building on this, we show that the regularization of such a QP yields an $\varepsilon$-approximated v-GNE with suboptimality vanishing linearly in the regularization parameter. Next, we propose an accelerated proximal-point scheme and an accelerated projected-gradient method, both attaining an $\mathcal O(1/k^2)$-approximated v-GNE at the $k$-th iteration. We also demonstrate that an invertible Jacobian of the game allows for reduction to a lower-dimensional QP. Theoretical analysis and numerical experiments show the proposed methods substantially outperform the existing approaches to solve monotone linear-quadratic v-GNE problems.

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arXiv2026

Solving Multiparametric Generalized Nash Equilibrium Problems and Explicit Game-Theoretic Model Predictive Control

We present a method for computing explicit solutions to parametric generalized Nash equilibrium (GNE) problems with convex quadratic cost functions and linear coupling and local constraints. Assuming that the parameters enter only the linear terms of the cost functions and the constraint right-hand sides, we provide the exact multiparametric solution of the GNE problem. Such a solution enables: (i) minimal real-time computation; (ii) inherent interpretability and explainability, as well as exact enumeration of all multiple equilibria; (iii) selection of desired GNE solution types in the case of infinitely many equilibria; and (iv) zero-shot updates of the GNE solution in response to changes in constraint right-hand sides and/or linear costs. In line with explicit model predictive control (MPC) approaches, we apply our method to solve game-theoretic MPC problems, also known as receding horizon games, explicitly. We compare its performance against centralized solvers in a battery charging game and a toy two-mass-spring-damper system control problem. A Python implementation of the algorithms presented in this paper is available at https://github.com/bemporad/nash_mpqp.

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arXiv2026

Learning practically stabilizing output-feedback nonlinear controllers

This paper addresses the problem of learning an output-feedback surrogate controller offline that approximates a given, possibly computationally expensive, nonlinear controller-observer pair. The surrogate is modeled as a recurrent dynamical system and is trained to imitate closed-loop input/output trajectories generated by the given controller. Beyond imitation accuracy, the offline training problem promotes input-to-state practical stability by incorporating estimated state trajectories to learn a candidate Lyapunov function. The approach is validated on a nonlinear continuous stirred tank reactor, where constraint satisfaction and practical stability are assessed through a probabilistic validation approach. The numerical results highlight the benefit of jointly learning the Lyapunov function by comparing against an imitation-only baseline.

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arXiv2026

Characterization and Computation of Feedback Nash Equilibria in Scalar Discounted N-Player Linear Quadratic Games

This paper studies feedback Nash equilibria (FNE) in scalar discounted linear quadratic (LQ) games with $N$ players. By explicitly incorporating the discount factor, we show that finite-cost equilibria may fail to stabilize the original system, motivating a distinction between FNE and stable FNE together with a sufficient stability condition. Based on a parametric characterization of the policies, we propose numerical methods for computing all equilibria. Particular attention is devoted to the symmetric game, where a closed-form expression of the symmetric FNE and conditions for the existence of up to $M\leq2^N-2$ equilibria are derived. Numerical experiments illustrate how equilibrium multiplicity depends on the game configuration and highlight the emergence of finite-cost non-stabilizing equilibria.

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arXiv2026

Amortized Nonlinear Model Predictive Control

Nonlinear Model Predictive Control requires solving a constrained nonlinear program (NLP) in real-time at every sampling instant, a computational bottleneck that limits deployment on resource-constrained hardware or at high sampling rates. We address this challenge for the broad class of input-affine nonlinear systems to show that the optimal control move can be approximated by a state-dependent quadratic program (QP) whose cost parameters depend on the current state and reference. We propose a single-network residual-corrector architecture: a state-dependent analytic baseline provides initial QP parameters, and the network learns only the corrections needed to match the full NLP solution; the QP is solved by a differentiable interior-point layer, guaranteeing constraint satisfaction for the first control action. The network is trained offline on data generated by an NLP solver using a hybrid loss that combines supervised imitation and KKT-residual penalties. We validate the approach on a three-link planar robotic arm with Cartesian end-effector tracking, demonstrating orders-of-magnitude speedup over the NLP solver while maintaining comparable tracking performance.

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arXiv2026

Towards A Goldfarb-Idnani Variant for Strongly Monotone Linear-Quadratic Games

We analyze a simple variant of the Goldfarb-Idnani (GI) dual active-set method for computing variational generalized Nash equilibria of strongly monotone N-player games with convex quadratic costs and shared affine inequality and equality constraints. We show that several properties of the GI algorithm are maintained in spite of having a possibly non-symmetric pseudogradient matrix in the joint KKT system of the game, although convergence to an existing equilibrium is not guaranteed as in the original algorithm. Our numerical results show that the method is potentially competitive with alternative state-of-the-art algorithms, including for computing solutions of game-theoretic linear model predictive control laws.

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arXiv2026

Active Learning MPC Objective Functions from Preferences

Designing the objective function in Model Predictive Control (MPC) is challenging when performance assessment criteria are available only from human judgment. We adopt a preference-based learning (PbL) approach to learn the MPC objective function from preferences over trajectory pairs. However, the real-world application of PbL is often restricted by the significant cost or limited availability of human preference queries. To address this, Active Learning (AL) strategies seek to improve sampling efficiency, reducing the labeling effort required to obtain a well-performing classifier. We present two AL strategies for learning the MPC objective function from human preferences over pairwise system trajectories: a pool-based strategy that selects trajectory pairs that are both uncertain under the current surrogate and diverse relative to previously labeled comparisons, and a query-synthesis strategy that incorporates new trajectories using the current surrogate-driven MPC. Numerical results show that the proposed strategies yield closed-loop behaviors that align more with the expressed preference using fewer number of queries compared to a random sampling approach.

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arXiv2026

Worst-case Nonlinear Regression with Error Bounds

We propose an active-learning method for nonlinear minimax regression. Given a nonlinear function that can be arbitrarily evaluated over a compact set, we fit a surrogate model, such as a feedforward neural network, by minimizing the maximum absolute approximation error. To handle the nonsmoothness of this worst-case loss, we introduce a smooth $L_\infty$ approximation that enables efficient gradient-based training. The training set is iteratively enriched by querying points of largest error via global optimization. We also derive constant and input-dependent worst-case error bounds over the entire input domain. The approach is validated on approximations of nonlinear functions and nonconvex sets, uncertain models of nonlinear dynamics, and explicit model predictive control laws. A Python library is available at https://github.com/bemporad/maxfit.

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arXiv2026

Rollout Then Optimize: A One-Step Newton Refinement of Learned Policies for Nonlinear Model Predictive Control

We propose a computationally efficient rollout-then-optimize method to improve a learned control policy at deployment time. A learned policy provides a nominal trajectory, which is refined online by a single Newton step implemented via a Riccati recursion within a model predictive control (MPC) scheme. This refinement combines model knowledge with the learned policy at minimal additional computational cost. We establish bounds on the approximation error of the learned policy relative to the MPC policy and show that one Newton step reduces the suboptimality of the learned rollout quadratically in the policy approximation error. The proposed controller is validated in simulation on a constrained trajectory-tracking task for a quadcopter with nonlinear dynamics. Results highlight that the Newton step significantly improves the learned policy, achieving performance close to a fully converged MPC solution while requiring roughly half of the computational time. The code is available at https://github.com/aghezz1/rl-riccati.

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arXiv2026

Parametric Nonconvex Optimization via Convex Surrogates

This paper presents a novel learning-based approach to construct a surrogate problem that approximates a given parametric nonconvex optimization problem. The surrogate function is designed to be the minimum of a finite set of functions, given by the composition of convex and monotonic terms, so that the surrogate problem can be solved directly through parallel convex optimization. As a proof of concept, numerical experiments on a nonconvex path tracking problem confirm the approximation quality of the proposed method.

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arXiv2026

Dual MPC for quasi-Linear Parameter Varying systems

We present a dual Model Predictive Control (MPC) framework for the simultaneous identification and control of quasi-Linear Parameter Varying (qLPV) systems. The framework is composed of an online estimator for the states and parameters of the qLPV system, and a controller that leverages the estimated model to compute inputs with a dual purpose: tracking a reference output while actively exciting the system to enhance parameter estimation. The core of this approach is a robust tube-based MPC scheme that exploits recent developments in polytopic geometry to guarantee recursive feasibility and stability in spite of model uncertainty. The effectiveness of the framework in achieving improved tracking performance while identifying a model of the system is demonstrated through a numerical example.

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arXiv2026

NashOpt - A Python Library for Computing Generalized Nash Equilibria

NashOpt is an open-source Python library for computing and designing generalized Nash equilibria (GNEs) in noncooperative games with shared constraints and real-valued decision variables. The library exploits the joint Karush-Kuhn-Tucker (KKT) conditions of all players to handle both general nonlinear GNEs and linear-quadratic games, including their variational versions. Nonlinear games are solved via nonlinear least-squares formulations, relying on JAX for automatic differentiation. Linear-quadratic GNEs are reformulated as mixed-integer linear programs, enabling efficient computation of multiple equilibria. The framework also supports inverse-game and Stackelberg game-design problems. The capabilities of NashOpt are demonstrated through several examples, including noncooperative game-theoretic control problems of linear quadratic regulation and model predictive control. The library is available at https://github.com/bemporad/nashopt

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arXiv2026

On Two-Player Scalar Discrete-Time Linear Quadratic Games

For the characterization of Feedback Nash Equilibria (FNE) in linear quadratic games, this paper provides a detailed analysis of the discrete-time discounted coupled best-response equations for the scalar two-player setting, together with a set of analytical tools for the classification of local saddle property for the iterative best-response method. Through analytical and numerical results we show the importance of classification, revealing an anti-coordination scheme in the case of multiple solutions. Particular attention is given to the symmetric case, where identical cost function parameters allow closed-form expressions and explicit necessary and sufficient conditions for the existence and multiplicity of FNE. We also present numerical results that illustrate the theoretical findings and offer foundational insights for the design and validation of iterative NE-seeking methods.

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arXiv2026

Active Learning-Based Input Design for Angle-Only Initial Relative Orbit Determination

Accurate relative orbit determination is a significant challenge in modern space operations, particularly when relying only on angular measurements. The inherent observability limitations of this approach make initial state estimation difficult, directly impacting mission safety and performance. This work proposes a hybrid estimation and control strategy for autonomous rendezvous. An active learning (AL) based algorithm designs the initial input control sequence by maximizing the exploration of the output space, thereby enhancing the observability of the initial relative state for the angle-only initial relative orbit determination (IROD) problem. The IROD solution provides a batch estimate of the initial relative state and its analytical covariance, which quantifies the estimation quality and determines the transition point to recursive filtering. Once the uncertainty is sufficiently low, an Extended Kalman Filter (EKF) is initialized with the IROD solution and takes over for sequential estimation, providing state estimates to a Model Predictive Controller (MPC) to complete the rendezvous. The proposed framework is validated through numerical simulations, demonstrating its ability to reliably resolve the scale ambiguity, outperform baseline excitation strategies, and successfully execute an end-to-end rendezvous from initial estimation to final approach.

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arXiv2025

Combined Learning of Linear Parameter-Varying Models and Robust Control Invariant Sets

Dynamical models identified from data are frequently employed in control system design. However, decoupling system identification from controller synthesis can result in situations where no suitable controller exists after a model has been identified. In this work, we introduce a novel control-oriented regularization in the identification procedure to ensure the existence of a controller that can enforce constraints on system variables robustly. The combined identification algorithm includes: (i) the concurrent learning of an uncertain model and a nominal model using an observer; (ii) a regularization term on the model parameters defined as the size of the largest robust control invariant set for the uncertain model. To make the learning problem tractable, we consider nonlinear models in quasi Linear Parameter-Varying (qLPV) form, utilizing a novel scheduling function parameterization that facilitates the derivation of an associated uncertain linear model. The robust control invariant set is represented as a polytope, and we adopt novel results from polytope geometry to derive the regularization function as the optimal value of a convex quadratic program. Additionally, we present new model-reduction approaches that exploit the chosen model structure. Numerical examples on classical identification benchmarks demonstrate the efficacy of our approach. A simple control scheme is also derived to provide an example of data-driven control of a constrained nonlinear system.

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