  In this paper, we extend the geometric descent method recently proposed by
Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite
problems. We prove that our proposed algorithm, dubbed geometric proximal
gradient method (GeoPG), converges with a linear rate $(1-1/\sqrt{\kappa})$ and
thus achieves the optimal rate among first-order methods, where $\kappa$ is the
condition number of the problem. Numerical results on linear regression and
logistic regression with elastic net regularization show that GeoPG compares
favorably with Nesterov's accelerated proximal gradient method, especially when
the problem is ill-conditioned.
