Optimizing second-order differential equation systems

Abstract

In this article we study some continuous versions of the Fletcher-Reeves iteration for minimization described by a system of second-order differential equations. This problem has been studied in earlier papers [19, 20] under the assumption that the minimizing function is strongly convex. Now instead of the strong convexity, only the convexity of the minimizing function will be required. We will use the Tikhonov regularization [28, 29] to obtain the minimal norm solution as the asymptotically stable limit point of the trajectories.

Keywords

• Fletcher-Reeves iteration
• second-order differential equation
• minimizing trajectory
• stationary point in limit
• Lyapunov-type methods