On the first- and second-order strongly monotone dynamical systems and minimization problems

Abstract

Motivated by application to Tikhonov regularization in convex minimization where the objective functions are strongly convex, we study the rate of convergence of the first- and second-order evolution equations associated with a maximal strongly monotone operator on a real Hilbert space. We show that the convergence rate of solutions to the second-order evolution equation of monotone type to a zero of the monotone operator (or a minimum point of a convex function) is faster than the first-order one when the maximal monotone operator is strongly monotone. Since the bounded solutions of the second-order evolution equation on the half line are not directly computable, we have used the computation of solutions to the corresponding second-order boundary value problem of monotone type. By using the numerical methods and approximation of solutions to the second-order boundary value problem associated with a gradient of a convex function with at least a minimum point, we approximate a minimum of the convex function. Finally, with a simple concrete example a comparison between this method and the steepest descent method is presented.