2015/03/16 by Radu Ioan Boţ, Bot, Radu Ioan, Ernö Robert Csetnek +1 · 1 citation
Computer Science · Mathematics · #34G25 #47H05 #47J25 #90C25 #Contact Mechanics and Variational Inequalities #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1503.04652
openalex publication_date 2015/03/16 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We begin by considering second order dynamical systems of the from nx(t) + \γ(t) x(t) + \λ(t)B(x(t))=0, where B: cal\nH\→ cal H is a cocoercive operator defined on a real Hilbert space\n cal H, \λ:[0,+\∞)\→ [0,+\∞) is a relaxation\nfunction and \γ:[0,+\∞)\→ [0,+\∞) a damping function,\nboth depending on time. For the generated trajectories, we show existence and\nuniqueness of the generated trajectories as well as their weak asymptotic\nconvergence to a zero of the operator B. The framework allows to address from\nsimilar perspectives second order dynamical systems associated with the problem\nof finding zeros of the sum of a maximally monotone operator and a cocoercive\none. This captures as particular case the minimization of the sum of a\nnonsmooth convex function with a smooth convex one. Furthermore, we prove that\nwhen B is the gradient of a smooth convex function the value of the latter\nconverges along the ergodic trajectory to its minimal value with a rate of\n cal O(1/t).\n