Tikhonov regularized mixed-order primal-dual dynamical system for convex optimization problems with linear equality constraints

Abstract: In Hilbert spaces, we consider a Tikhonov regularized mixed-order primal-dual dynamical system for a convex optimization problem with linear equality constraints. The dynamical system with general time-dependent parameters: viscous damping and temporal scaling can derive certain existing systems when special parameters are selected. When these parameters satisfy appropriate conditions and the Tikhonov regularization parameter \epsilon(t) approaches zero at an appropriate rate, we analyze the asymptotic convergence properties of the proposed system by constructing suitable Lyapunov functions. And we obtain that the objective function error enjoys O(1/(t^2\beta(t))) convergence rate. Under suitable conditions, it can be better than O(1/(t^2)). In addition, we utilize the Lyapunov analysis method to obtain the strong convergence of the trajectory generated by the Tikhonov regularized dynamical system. In particular, when Tikhonov regularization parameter \epsilon(t) vanishes to 0 at some suitable rate, the convergence rate of the primal-dual gap can be o(1/(\beta(t))). The effectiveness of our theoretical results has been demonstrated through numerical experiments.