  In this paper we propose a primal-dual proximal extragradient algorithm to
solve the generalized Dantzig selector (GDS) estimation problem, based on a new
convex-concave saddle-point (SP) reformulation. Our new formulation makes it
possible to adopt recent developments in saddle-point optimization, to achieve
the optimal $O(1/k)$ rate of convergence. Compared to the optimal non-SP
algorithms, ours do not require specification of sensitive parameters that
affect algorithm performance or solution quality. We also provide a new
analysis showing a possibility of local acceleration to achieve the rate of
$O(1/k^2)$ in special cases even without strong convexity or strong smoothness.
As an application, we propose a GDS equipped with the ordered $\ell_1$-norm,
showing its false discovery rate control properties in variable selection.
Algorithm performance is compared between ours and other alternatives,
including the linearized ADMM, Nesterov's smoothing, Nemirovski's mirror-prox,
and the accelerated hybrid proximal extragradient techniques.
