Fast Continuous Dynamics Inside the Graph of Subdifferentials of Nonsmooth Convex Functions

In a Hilbert framework, we introduce a new class of second-order dynamical systems that combine viscous and geometric damping but also a time rescaling process for nonsmooth convex minimization. A main feature of these systems is to produce trajectories that lie in the graph of the Fenchel subdiffer...

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Bibliographic Details
Published in:Applied mathematics & optimization 2024-02, Vol.89 (1), p.1, Article 1
Main Authors: Maingé, Paul-Emile, Weng-Law, André
Format: Article
Language:English
Subjects:
Algorithms
Calculus of Variations and Optimal Control
Optimization
Control
Convergence
Convex analysis
Damping
Differential equations
Dynamical systems
Hilbert space
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Optimization
Regularization
Rescaling
Simulation
Systems Theory
Theoretical
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Summary:In a Hilbert framework, we introduce a new class of second-order dynamical systems that combine viscous and geometric damping but also a time rescaling process for nonsmooth convex minimization. A main feature of these systems is to produce trajectories that lie in the graph of the Fenchel subdifferential of the objective. Moreover, they do not incorporate any regularization or smoothing processes. This new class originates from some combination of a continuous Nesterov-like dynamic and the Minty representation of subdifferentials. These models are investigated through first-order reformulations that amount to dynamics involving three variables: two solution trajectories (including an auxiliary one) and another one associated with subgradients. We prove the weak convergence towards equilibria for the solution trajectories, as well as properties of fast convergence to zero for their velocities. Remarkable convergence rates (possibly of exponential-type) are also established for the function values. We additionally state notable properties of fast convergence to zero for the subgradients trajectory and for its velocity. Some numerical experiments are performed so as to illustrate the efficiency of our approach. The proposed models offer a new and well-adapted framework for discrete counterparts, especially for structured minimization problems. Inertial algorithms with a correction term are then suggested relative to this latter context.
ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-023-10055-9