A second order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities

摘要

We consider the minimization of a convex objective function subject to theset of minima of another convex function, under the assumption that bothfunctions are twice continuously differentiable. We approach this optimizationproblem from a continuous perspective by means of a second order dynamicalsystem with Hessian-driven damping and a penalty term corresponding to theconstrained function. By constructing appropriate energy functionals, we proveweak convergence of the trajectories generated by this differential equation toa minimizer of the optimization problem as well as convergence for theobjective function values along the trajectories. The performed investigationsrely on Lyapunov analysis in combination with the continuous version of theOpial Lemma. In case the objective function is strongly convex, we can evenshow strong convergence of the trajectories.