  We propose a framework to use Nesterov's accelerated method for constrained
convex optimization problems. Our approach consists of first reformulating the
original problem as an unconstrained optimization problem using a continuously
differentiable exact penalty function. This reformulation is based on replacing
the Lagrange multipliers in the augmented Lagrangian of the original problem by
Lagrange multiplier functions. The expressions of these Lagrange multiplier
functions, which depend upon the gradients of the objective function and the
constraints, can make the unconstrained penalty function non-convex in general
even if the original problem is convex. We establish sufficient conditions on
the objective function and the constraints of the original problem under which
the unconstrained penalty function is convex. This enables us to use Nesterov's
accelerated gradient method for unconstrained convex optimization and achieve a
guaranteed rate of convergence which is better than the state-of-the-art
first-order algorithms for constrained convex optimization. Simulations
illustrate our results.
