Abstract
In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with asymptotically vanishing damping term. The system is formulated in terms of the augmented Lagrangian associated to the minimization problem. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. In case the objective function has Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically weakly converges to a primal-dual optimal solution of the underlying minimization problem. To the best of our knowledge, this is the first result which guarantees the convergence of the trajectory generated by a primal-dual dynamical system with asymptotic vanishing damping. Moreover, we will rediscover in case of the unconstrained minimization of a convex differentiable function with Lipschitz continuous gradient all convergence statements obtained in the literature for Nesterov's accelerated gradient method.
Highlights
The augmented Lagrangian Method (ALM) [49], the Alternating Direction Method of Multipliers (ADMM) [35, 32] and some of their variants have proved to be very suitable when solving large-scale structured convex optimization problems
Since the primal-dual systems of optimality conditions to be solved can be equivalently formulated as monotone inclusion problems, see [48, 49, 50], the above-mentioned methods are intimately linked with numerical algorithms designed to find a zero of a maximally monotone operator
This close connection has been used in recent works addressing the acceleration of ADMM/ALM methods via inertial dynamics
Summary
In this paper we will deal with the optimization problem min f pxq , subject to Ax “ b (1.1). The interplay between continuous-time dissipative dynamical systems and numerical algorithms for solving optimization problems has been subject of an intense research activity. Since the primal-dual systems of optimality conditions to be solved can be equivalently formulated as monotone inclusion problems, see [48, 49, 50], the above-mentioned methods are intimately linked with numerical algorithms designed to find a zero of a maximally monotone operator. This close connection has been used in recent works addressing the acceleration of ADMM/ALM methods via inertial dynamics. Continuous-time approaches for structured convex minimization problems formulated in the spirit of the full splitting paradigm have been recently addressed in [31] and, closely connected to our approach, in [56, 37, 11], to which we will have a closer look in Subsection 2.3
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