  We propose an unconstrained optimization method based on the well-known
primal-dual hybrid gradient (PDHG) algorithm. We first formulate the optimality
condition of the unconstrained optimization problem as a saddle point problem.
We then compute the minimizer by applying generalized primal-dual hybrid
gradient algorithms. Theoretically, we demonstrate the continuous-time limit of
the proposed algorithm forms a class of second-order differential equations,
which contains and extends the heavy ball ODEs and Hessian-driven damping
dynamics. Following the Lyapunov analysis of the ODE system, we prove the
linear convergence of the algorithm for strongly convex functions.
Experimentally, we showcase the advantage of algorithms on several convex and
non-convex optimization problems by comparing the performance with other
well-known algorithms, such as Nesterov's accelerated gradient methods. In
particular, we demonstrate that our algorithm is efficient in training
two-layer and convolution neural networks in supervised learning problems.
