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- Title
Fast Continuous Dynamics Inside the Graph of Subdifferentials of Nonsmooth Convex Functions.
- Authors
Maingé, Paul-Emile; Weng-Law, André
- Abstract
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.
- Subjects
SUBDIFFERENTIALS; DYNAMICAL systems; DIFFERENTIAL equations; VELOCITY; NONSMOOTH optimization
- Publication
Applied Mathematics & Optimization, 2024, Vol 89, Issue 1, p1
- ISSN
0095-4616
- Publication type
Article
- DOI
10.1007/s00245-023-10055-9