  The Condat-V\~u algorithm is a widely used primal-dual method for optimizing
composite objectives of three functions. Several algorithms for optimizing
composite objectives of two functions are special cases of Condat-V\~u,
including proximal gradient descent (PGD). It is well-known that PGD exhibits
suboptimal performance, and a simple adjustment to PGD can accelerate its
convergence rate from $\mathcal{O}(1/T)$ to $\mathcal{O}(1/T^2)$ on convex
objectives, and this accelerated rate is optimal. In this work, we show that a
simple adjustment to the Condat-V\~u algorithm allows it to recover accelerated
PGD (APGD) as a special case, instead of PGD. We prove that this accelerated
Condat--V\~u algorithm achieves optimal convergence rates and significantly
outperforms the traditional Condat-V\~u algorithm in regimes where the
Condat--V\~u algorithm approximates the dynamics of PGD. We demonstrate the
effectiveness of our approach in various applications in machine learning and
computational imaging.
