1. A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities
- Author
-
Radu Ioan Boţ and Ernö Robert Csetnek
- Subjects
TheoryofComputation_MISCELLANEOUS ,Lyapunov function ,Hessian matrix ,convex optimization ,Control and Optimization ,Dynamical systems theory ,Lyapunov analysis ,MathematicsofComputing_NUMERICALANALYSIS ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Article ,symbols.namesake ,Dynamical systems ,Applied mathematics ,0101 mathematics ,Mathematics ,021103 operations research ,Weak convergence ,Applied Mathematics ,Newton dynamics ,nonautonomous systems ,010101 applied mathematics ,Maxima and minima ,Convex optimization ,Variational inequality ,symbols ,Convex function - Abstract
We consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second-order dynamical system with Hessian-driven damping and a penalty term corresponding to the constrained function. By constructing appropriate energy functionals, we prove weak convergence of the trajectories generated by this differential equation to a minimizer of the optimization problem as well as convergence for the objective function values along the trajectories. The performed investigations rely on Lyapunov analysis in combination with the continuous version of the Opial Lemma. In case the objective function is strongly convex, we can even show strong convergence of the trajectories.
- Published
- 2018