An inertial system with Hessian-driven damping is developed, which complements existing results on smooth saddle point problems with bilinear coupling and considers the affinely constrained convex optimization problem.
Abstract
Featuring Hessian-driven damping, two inertial primal dual dynamical systems are proposed for solving smooth saddle point problems with bilinear coupling. For convex-concave functions, we establish a convergence rate $\mathcal{O}\left( \frac{1}{t^2} \right)$ for the primal dual gap; for strongly convex-strongly concave functions, we obtain an asymptotic rate $\mathcal{O}\left( \frac{1}{t^{\alpha-1}} \right)$ ($\alpha\ge 3$ is the damping parameter) without knowledge of the strong convexity parameters, and an accelerated linear convergence rate when the strong convexity parameters are known. As an application of the proposed inertial systems, we also consider the affinely constrained convex optimization problem, and develop an inertial system with Hessian-driven damping, which complements existing results.
This work considers convex optimization with nonlinear inequality constraints and develops a primal-dual multiplier framework that is consistent in continuous and discrete time and is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.
The strong convergence of the primal trajectory is proved and fast convergence rates under the same parameter assumptions are derived under the same parameter assumptions, without imposing any eventual inside/outside-ball condition.
This paper investigates inertial primal-dual dynamics with implicit Hessian-driven damping for strongly convex optimization problems with linear equality constraints with fast convergence rates and derives an inertial accelerated primal-dual algorithm for solving the strongly convex optimization problems.
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