  We study a new penalty reformulation of constrained convex optimization based
on the softplus penalty function. We develop novel and tight upper bounds on
the objective value gap and the violation of constraints for the solutions to
the penalty reformulations by analyzing the solution path of the reformulation
with respect to the smoothness parameter. We use these upper bounds to analyze
the complexity of applying gradient methods, which are advantageous when the
number of constraints is large, to the reformulation.
