Second-order dynamical systems associated to variational inequalities

作者：Bot Radu Ioan^*; Csetnek Erno Robert

来源：APPLICABLE ANALYSIS, 2017, 96(5): 799-809.

DOI：10.1080/00036811.2016.1157589

• Dynamical systems
• Lyapunov analysis
• convex optimization
• nonautonomous systems
• 34G25
• 47J25
• 47H05
• 90C25

We investigate the asymptotic convergence of the trajectories generated by the second-order dynamical system x(t) + x(t) +. f(x(t)) + (t) (x(t)) = 0, where f,. : H. R are the convex and smooth functions defined on a real Hilbert space H, > 0 and is a function of time which controls the penalty term. We show weak convergence of the trajectories to a minimizer of the function f over the (nonempty) set of minima of. as well as convergence for the objective function values along the trajectories, provided a condition expressed via the Fenchel conjugate of. is fulfilled. When the function f is assumed to be strongly convex, we can even show strong convergence of the trajectories. The results can be seen as the second-order counterparts of the ones given by Attouch and Czarnecki (Journal of Differential Equations 248(6), 1315-1344, 2010) for first-order dynamical systems associated to the constrained variational inequalities. At the same time we give a positive answer to an open problem posed by Attouch and Czarnecki in a recent preprint.

• 出版日期2017