A fast primal-dual algorithm via dynamical system with variable mass for linearly constrained convex optimization

We aim to solve the linearly constrained convex optimization problem whose objective function is the sum of a differentiable function and a non-differentiable function. We first propose an inertial continuous primal-dual dynamical system with variable mass for linearly constrained convex optimization problems with differentiable objective functions. The dynamical system is composed of a second-order differential equation with variable mass for the primal variable and a first-order differential equation for the dual variable. The fast convergence properties of the proposed dynamical system are proved by constructing a proper energy function. We then extend the results to the case where the objective function is non-differentiable, and a new accelerated primal-dual algorithm is presented. When both variable mass and time scaling satisfy certain conditions, it is proved that our new algorithm owns fast convergence rates for the objective function residual and the feasibility violation. Some preliminary numerical results on the \(\ell _{1}\) ${\ell }_1$ – \(\ell _{2}\) ${\ell }_2$ minimization problem demonstrate the validity of our algorithm.