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Convergent Second-Order Methods for Min-Max Optimizations

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Convergent Second-Order Methods for Min-Max Optimizations

American control Conference (ACC)

作者： Raphael Chinchilla Guosong Yang João Hespanha Center for Control Dynamical Systems and Computation University of California Santa Barbara United States of America

We address the use of second-order methods to solve optimizations of the form\begin{equation*}\mathop {\min }\limits_{u \in \mathcal{U}} \mathop {\max }\limits_{d \in \mathcal{D}} f(u,d),\tagcontrol\end{equation*}for a twice continuously differentiable function $f:\mathcal{U} \times \mathcal{D} \to \mathbb{R}$ and sets $\mathcal{U} \subset {\mathbb{R}^{{n_u}}},\mathcal{D} \subset {\mathbb{R}^{{n_d}}}$. This type of optimization arises in numerous applications, including robust machine learning [1], model predictive control [2], [3], and in reformulating stochastic programming as a min-max optimizations [4], [5].When the sets $\mathcal{U}$ and $\mathcal{D}$ are compact and convex and the function f (u, d) is convex with respect to u and concave with respect to d, the min and max in (1) commute [6] and the optimization becomes relatively simple. However, we are especially interested here in problems for which such assumptions do not hold, the min and max do not commute, and for which the optimizations may have local optima that are not *** this talk, we address the design of algorithms to solve nonconvex-nonconcave min-max optimizations like (1) using second order methods. These algorithms modify the Hessian matrix to obtain a search direction that can be seen as the solution to a quadratic program that locally approximates the min-max problem. We show that by selecting this type of modification appropriately, the only stable points of the resulting iterations are local min-max points. For min-max model predictive control problems, these algorithms leads to computation times that scale linearly with the horizon *** more information please see the main tutorial paper [7].

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Learning Residual Dynamics via Physics-Augmented Neural Networks: Application to Vapor Compression Cycles

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Learning Residual Dynamics via Physics-Augmented Neural Netw...

American control Conference (ACC)

作者： Raphael Chinchilla Vedang M. Deshpande Ankush Chakrabarty Christopher R. Laughman Center for Control Dynamical Systems and Computation University of California Santa Barbara Mitsubishi Electric Research Laboratories

In order to improve the control performance of vapor compression cycles (VCCs), it is often necessary to construct accurate dynamical models of the underlying thermo-fluid dynamics. These dynamics are represented by complex mathematical models that are composed of large systems of nonlinear and numerically stiff differential algebraic equations (DAEs). The effects of nonlinearity and stiffness may be ameliorated by using physics-based models to describe characteristic system behaviors, and approximating the residual (unmodeled) dynamics using neural networks. In these so-called ‘physics-augmented’ or ‘physics-informed’ machine learning approaches, the learning problem is often solved by jointly estimating parameters of the physics component model and weights of the network. Furthermore, such approaches also often assume the availability of full-state information, which typically are not available in practice for energy systems such as VCCs after deployment. Rather than concurrently performing state/parameter estimation and network training, which often leads to numerical instabilities, we propose a framework for decoupling the network training from the joint state/parameter estimation problem by employing state-constrained Kalman smoothers customized for VCC applications. We show the effectiveness of our proposed framework on a Julia-based, high-fidelity simulation environment calibrated to a model of a commercially-available VCC and achieve an accuracy of 98% calculated over 24 states and multiple initial conditions under realistic operating conditions.

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Proximal Gradient Dynamics and Feedback control for Equality-Constrained Composite Optimization

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arXiv 2025年

作者： Centorrino, Veronica Rossi, Francesca Bullo, Francesco Russo, Giovanni DIEM University of Salerno Italy Scuola Superiore Meridionale Italy Center for Control Dynamical Systems and Computation UC Santa Barbara CA United States

This paper studies equality-constrained composite minimization problems. This class of problems, capturing regularization terms and convex inequality constraints, naturally arises in a wide range of engineering and machine learning applications. To tackle these minimization problems, we introduce the proportional–integral proximal gradient dynamics (PI–PGD): a closed-loop system where the Lagrange multipliers are control inputs and states are the problem decision variables. First, we establish the equivalence between the minima of the optimization problem and the equilibria of the PI–PGD. Then, leveraging tools from contraction theory, we give a comprehensive convergence analysis for the dynamics, showing linear–exponential convergence towards the equilibrium. That is, the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. Our findings are illustrated numerically on a set of representative examples, which include an application to entropic-regularized optimal transport. Copyright © 2025, The Authors. All rights reserved.

关键词： Lagrange multipliers

Contractivity of Distributed Optimization and Nash Seeking Dynamics

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arXiv 2023年

作者： Gokhale, Anand Davydov, Alexander Bullo, Francesco The Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93106 United States

In this letter, we study distributed optimization and Nash equilibrium-seeking dynamics from a contraction theoretic perspective. Our first result is a novel bound on the logarithmic norm of saddle matrices. Second, for distributed gradient flows based upon incidence and Laplacian constraints over arbitrary topologies, we establish strong contractivity over an appropriate invariant vector subspace. Third, we give sufficient conditions for strong contractivity in pseudogradient and best response games with complete information, show the equivalence of these conditions, and consider the special case of aggregative games. Copyright © 2023, The Authors. All rights reserved.

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A Stochastic Surveillance Stackelberg Game: Co-Optimizing Defense Placement and Patrol Strategy

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IEEE Transactions on Automatic control 2025年

作者： John, Yohan Díaz-García, Gilberto Duan, Xiaoming Marden, Jason R. Bullo, Francesco UC Santa Barbara Center for Control Dynamical Systems and Computation Santa BarbaraCA93106-5070 United States Shanghai Jiao Tong University Department of Automation Shanghai200240 China

Stochastic patrol routing is known to be advantageous in adversarial settings;however, the optimal choice of stochastic routing strategy is dependent on a model of the adversary. We adopt a worst-case omniscient adversary model from the literature and extend the formulation to accommodate heterogeneous defenses at the various nodes of the graph. Introducing this heterogeneity leads to interesting new patrol strategies. We identify efficient methods for computing these strategies in certain classes of graphs. We assess the effectiveness of these strategies via comparison to an upper bound on the value of the game. Finally, we leverage the heterogeneous defense formulation to develop novel defense placement algorithms that complement the patrol strategies. © 1963-2012 IEEE.

关键词： Markov processes

RoSSO: A High-Performance Python Package for Robotic Surveillance Strategy Optimization Using JAX

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arXiv 2023年

作者： John, Yohan Hughes, Connor Díaz-García, Gilberto Marden, Jason R. Bullo, Francesco The Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93106-5070 United States

To enable the computation of effective randomized patrol routes for single- or multi-robot teams, we present RoSSO, a Python package designed for solving Markov chain optimization problems. We exploit machine-learning techniques such as reverse-mode automatic differentiation and constraint parametrization to achieve superior efficiency compared to general-purpose nonlinear programming solvers. Additionally, we supplement a game-theoretic stochastic surveillance formulation in the literature with a novel greedy algorithm and multi-robot extension. We close with numerical results for a police district in downtown San Francisco that demonstrate RoSSO’s capabilities on our new formulations and the prior work. Copyright © 2023, The Authors. All rights reserved.

关键词： Python

computation of Hopf Bifurcation Points in the Magnetic Levitation System 4th

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Computation of Hopf Bifurcation Points in the Magnetic Levit...

4th International Conference on Information and Communication Technology for Intelligent systems, ICTIS 2020

作者： Valluru, Sudarshan K. Gupta, Anshul Verma, Aditya Center for Control of Dynamical Systems and Computation Department of Electrical Engineering Delhi Technological University Delhi110042 India

ISBN: (纸本)9789811570773

This research brief aims to study the bifurcation analysis of an electromagnetic levitation (maglev) system. The bifurcation analysis involves first performing the numerical analysis, and then the simulations using MATLAB are analyzed to verify the results. The study requires derivation of a mathematical model for the maglev system, which can be used for the bifurcation analysis. The bifurcation analysis involves a numerical comparison of the system equations with the predefined constraints. The input voltage is being used as the bifurcation parameter, which is by ball’s position, ball’s velocity, and the current flowing through the circuit. The analysis allows us to compute a combination of values of parameters, when simulations are performed to check system characteristics, periodic oscillations at these values are obtained. Hence, it proves the existence of Hopf bifurcation at the computed value of system parameters. © 2021, The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

关键词： Hopf bifurcation

Inference of Infrastructure Network Flows via Physics-Inspired Implicit Neural Networks

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Inference of Infrastructure Network Flows via Physics-Inspir...

control Technology and Applications (CCTA),

作者： Francesco Seccamonte Ambuj K. Singh Francesco Bullo Center for Control Dynamical Systems and Computation University of California Santa Barbara Department of Computer Science University of California Santa Barbara

We study the problem of inferring edge flows and nodal injections in infrastructure networks. Leveraging the Thomson’s Principle from the electric circuits literature, we setup a framework to jointly learn network parameters and missing states. Despite being application agnostic, the proposed approach captures the fundamental physics of the infrastructure, and is able to handle partial observation, node and edge features as well as operational constraints. The physics inspired learning framework leads to a bilevel optimization problem, which is NP hard in general. By exploiting convexity properties, we reformulate the problem as a single level optimization, composed of a graph neural network and an additional implicit layer. The resulting architecture can be trained using standard gradient-based methods. We assess the validity of the proposed approach on two different infrastructure networks (power and traffic), and show it outperforms the current state of the art.

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Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees

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arXiv 2024年

作者： Jaffe, Sean Davydov, Alexander Lapsekili, Deniz Singh, Ambuj K. Bullo, Francesco Center for Control Dynamical Systems and Computation University of California Santa Barbara United States Department of Computer Science University of California Santa Barbara United States

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. We present Extended Linearized Contracting Dynamics (ELCD), the first neural network-based dynamical system with global contractivity guarantees in arbitrary metrics. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable, (ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets. © 2024, CC BY.

关键词： Linearization