2021/06/23 by Radu Ioan Boţ, Bot, Radu Ioan, Dang-Khoa Nguyen +1 · 4 citations
Engineering · Mathematics · Medicine · #37N40 #46N10 #65K10 #90C25 #Bone and Joint Diseases #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.2106.12294
openalex publication_date 2021/06/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this work, we approach the minimization of a continuously differentiable\nconvex function under linear equality constraints by a second-order dynamical\nsystem with asymptotically vanishing damping term. The system is formulated in\nterms of the augmented Lagrangian associated to the minimization problem. We\nshow fast convergence of the primal-dual gap, the feasibility measure, and the\nobjective function value along the generated trajectories. In case the\nobjective function has Lipschitz continuous gradient, we show that the\nprimal-dual trajectory asymptotically weakly converges to a primal-dual optimal\nsolution of the underlying minimization problem. To the best of our knowledge,\nthis is the first result which guarantees the convergence of the trajectory\ngenerated by a primal-dual dynamical system with asymptotic vanishing damping.\nMoreover, we will rediscover in case of the unconstrained minimization of a\nconvex differentiable function with Lipschitz continuous gradient all\nconvergence statements obtained in the literature for Nesterov's accelerated\ngradient method.\n