Discrete-time Euler-smoothing methods for time-varying convex constrained optimization

Abstract

In this paper, we propose two kinds of prediction–correction methods, discrete-time Euler-smoothing methods, for solving the general time-varying convex constrained optimization problem. By using the complementarity function and smoothing technique, the Karush-Kuhn–Tucker (KKT) trajectory of the problem can be reformulated as a system of smooth equations and considering the dynamics of the system of smooth equations in a discrete constant time-sampling periods scheme. The proposed methods mainly consist of the prediction step derived by Euler approximation and the correction step generated by one or multiple Newton iterations. Under some reasonable conditions, the approximate solution trajectories produced by the methods achieve the asymptotic error bound behaving as O(h4) with respect to optimal trajectory both in the primal and dual space as the smoothing parameter approaches to zero, where h is the sampling period. Finally, the experimental results show that our methods are efficient, and the steady-state tracking errors are much smaller than the other prediction–correction methods under the same conditions.