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Article
Mathematics, Applied
Giacomo Borghi et al.
Summary: In this work, a numerical method for high-dimensional constrained nonlinear optimization problems using particle-based gradient-free techniques is developed. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced. The method reformulates the constrained problem into an unconstrained one and extends to constrained settings. An iterative strategy is proposed to update the penalty parameter based on the constrained violation. Convergence of the proposed method is shown using a mean-field description of the CBO dynamics, and the properties of the algorithm are analyzed. Numerical examples demonstrate the theoretical findings and the good performance of the new method, even in high dimensions.
SIAM JOURNAL ON OPTIMIZATION
(2023)
Article
Automation & Control Systems
Gennaro Ciampa et al.
Summary: We prove the existence of Lipschitz-in-space optimal controls for a class of mean-field control problems with non-local continuity equation dynamics. The proof is based on a vanishing viscosity method, where we show the convergence of the problem with the addition of a diffusion term and a small viscosity parameter. By employing stochastic optimal control, we demonstrate the existence of an optimal control sequence for the problem with diffusion. We then construct the optimizer of the original problem by taking the viscosity parameter to zero.
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
(2023)
Article
Mathematics, Applied
Gianluca Orlando
Summary: We study a multi-agent system to model maritime crime, involving three populations of ships: commercial, pirate, and coast guard. We prove the well-posedness of the model and the existence of optimal controls. We then derive a mean-field PDE/PDE/ODE model for large numbers of ships and study the limit of the corresponding optimal control problems.
ADVANCES IN CONTINUOUS AND DISCRETE MODELS
(2023)
Article
Computer Science, Artificial Intelligence
Fabrizia Auletta et al.
Summary: This article proposes a simple yet effective set of local control rules for a small group of herder agents to collect and contain a large ensemble of non-cooperative, non-flocking stochastic target agents in the plane. The robustness of the proposed strategies to variations in the number of target agents and the strength of the repulsive force they experience when near the herders is investigated. The effectiveness of the proposed approach is confirmed through simulations in ROS and experiments on real robots.
Article
Mathematics, Applied
Giacomo Albi et al.
Summary: This paper studies a mean-field selective optimal control problem of multipopulation dynamics via transient leadership. The problem considers the spatial position and probability of belonging to a certain population for the agents, and introduces an activation function to tune the control on each agent. By using Gamma-convergence, the mean-field limit of the finite-particle control problem is identified, ensuring the convergence of optimal controls. Specific applications in the context of opinion dynamics are discussed.
APPLIED MATHEMATICS AND OPTIMIZATION
(2022)
Article
Mathematics
Giulia Cavagnari et al.
Summary: This paper focuses on the study of multi-agent deterministic optimal control problems. The Lagrangian, Eulerian, and Kantorovich formulations of the problems, as well as their relaxations, are thoroughly analyzed. Equivalence results among the different representations are presented, and the respective value functions are compared. Techniques and ideas from optimal transportation, control theory, Young measures, and evolution equations in Banach spaces are combined to achieve these goals. Additionally, consistency results are obtained as the number of particles/agents tends to infinity by exploiting the connections between Lagrangian and Eulerian descriptions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Benoît Bonnet et al.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics, Applied
Massimo Fornasier et al.
Summary: This paper addresses the global minimization problem of a possibly non-smooth and nonconvex objective function constrained on the unit hypersphere. It proposes a multi-agent derivative-free method based on consensus-based optimization. The proposed algorithm utilizes a drift towards an instantaneous consensus point to guide agents' movement on the sphere, and introduces an anisotropic random vector field to enhance exploration. The main results include the convergence proof of the numerical scheme to global minimizers under certain conditions. The introduction of the anisotropic stochastic term distinguishes this work from previous ones, enabling the algorithm to be dimension-independent and applicable in high-dimensional spaces. Numerical experiments demonstrate the effectiveness and versatility of the proposed algorithm compared to previous formulations with isotropic stochastic noise.
SIAM JOURNAL ON OPTIMIZATION
(2022)
Article
Mathematics
Benoit Bonnet et al.
Summary: In this paper, a general framework for studying differential inclusions in the Wasserstein space of probability measures is proposed, with solutions defined as absolutely continuous curves driven by measurable selection functions taking values in the space of vector fields. The foundational results of Filippov's theorem, Relaxation theorem, and compactness of solution sets are proved in this setting, based on novel estimates on solutions of continuity equations. These contributions are then applied to derive a new existence result for fully non-linear mean-field optimal control problems with closed-loop controls.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Luigi Ambrosio et al.
Summary: In this study, a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by replicator dynamics is introduced and examined. The evolution is described using Lagrangian and Eulerian descriptions, with the equivalence, existence, uniqueness, and stability of the solution proven. As a result of the stability analysis, convergence of the finite agents model to the mean-field formulation is obtained as the number of players approaches infinity.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Stefano Almi et al.
Summary: A multi-step Lagrangian scheme is proposed to approximate a nonlinear continuity equation arising from spatially inhomogeneous evolutionary games, which describes the evolution of spatially distributed agents with strategies, or labels, whose payoff depends on the current position of the agents. The scheme follows two stages: updating the distribution of strategies or labels based on a best performance criterion, and then using this to evolve the position of the agents. It provides a general convergence result in the space of probability measures, and also considers variants in special cases such as replicator-type systems and reversible Markov chains.
JOURNAL OF EVOLUTION EQUATIONS
(2021)
Article
Mathematics, Applied
Benoit Bonnet et al.
Summary: This article derives first-order necessary optimality conditions for a constrained optimal control problem in the Wasserstein space of probability measures, introduces a new notion of localised metric subdifferential, and investigates intrinsic linearised Cauchy problems associated with non-local continuity equations. By using these concepts, the Pontryagin Maximum Principle for optimal control problems with inequality final-point constraints is proven synthetically and geometrically, and sufficient conditions for the normality of the maximum principle are proposed.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Fabio Camilli et al.
Summary: This paper studies the numerical approximation of a system of PDEs from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The Caputo derivatives in the system are approximated using L1 schemes and the Hamiltonian is approximated using finite differences. The scheme for the Fokker-Planck equation is constructed to preserve the duality structure of the PDE system on the discrete level, and the well-posedness of the scheme and convergence to the solution of the continuous problem are proven.
JOURNAL OF DYNAMICS AND GAMES
(2021)
Article
Automation & Control Systems
Martin Burger et al.
Summary: A framework for computing optimal controls for problems with states in the space of probability measures was derived. New calculus was used to derive the first-order optimality system and establish a link between the adjoint in the space of probability measures and L-2 calculus in numerical simulations. Convergence rate of optimal controls from particle formulation to mean-field problem was proven as the number of particles tends to infinity.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Automation & Control Systems
Martin Burger et al.
Summary: This study proposes a mean-field optimal control problem for parameter identification of a given pattern, using the Wasserstein distance between probability measures as the cost functional. By deriving first-order optimality conditions and proposing a gradient descent method, relevant parameters such as rotation angle and force scaling can be identified, with a convergence rate proven for the controls as the number of particles used for discretization tends to infinity. The numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
(2021)
Article
Mathematics, Applied
Fabio Camilli et al.
Summary: The paper analyzes the properties of Measure Differential Equations (MDE) and highlights their connection with nonlocal continuity equations. It proves a representation result similar to the Superposition Principle by Ambrosio-Gigli-Savare, and provides alternative schemes converging to a solution of MDE, focusing on uniqueness/non-uniqueness phenomena.
KINETIC AND RELATED MODELS
(2021)
Article
Automation & Control Systems
Dante Kalise et al.
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
(2020)
Article
Computer Science, Interdisciplinary Applications
Martin Burger et al.
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Mathematics, Applied
Jose A. Carrillo et al.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2020)
Article
Mathematics, Applied
Benedetto Piccoli et al.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2020)
Article
Mathematical & Computational Biology
Claudia Totzeck et al.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2020)
Article
Physics, Fluids & Plasmas
Giacomo Dimarco et al.
Article
Mathematics, Applied
M. Fornasier et al.
EUROPEAN JOURNAL OF APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Benoit Bonnet et al.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2019)
Article
Mathematics, Interdisciplinary Applications
Andrea Tosin et al.
MULTISCALE MODELING & SIMULATION
(2019)
Article
Robotics
Alyssa Pierson et al.
IEEE TRANSACTIONS ON ROBOTICS
(2018)
Article
Mathematics, Applied
Jose A. Carrillo et al.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2018)
Article
Mathematics, Applied
Mattia Bongini et al.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2017)
Article
Mathematics, Applied
Giuseppe Maria Coclite et al.
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
(2017)
Article
Automation & Control Systems
Maria Laura Delle Monache et al.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2017)
Article
Mathematics, Applied
Giacomo Albi et al.
APPLIED MATHEMATICS AND OPTIMIZATION
(2017)
Article
Mathematics, Applied
Benedetto Piccoli et al.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2016)
Article
Mathematics, Applied
Giacomo Albi et al.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2016)
Article
Mathematics, Applied
Alessio Porretta
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2015)
Article
Statistics & Probability
Nicolas Fournier et al.
PROBABILITY THEORY AND RELATED FIELDS
(2015)
Article
Mathematics, Applied
Benedetto Piccoli et al.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2014)
Article
Mathematics, Applied
Luigi Ambrosio et al.
Article
Mathematics, Applied
Francois Bolley et al.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2011)
Article
Mathematics
V. I. Bogachev et al.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2007)
