  Line search (or backtracking) procedures have been widely employed into
first-order methods for solving convex optimization problems, especially those
with unknown problem parameters (e.g., Lipschitz constant). In this paper, we
show that line search is superfluous in attaining the optimal rate of
convergence for solving a convex optimization problem whose parameters are not
given a priori. In particular, we present a novel accelerated gradient descent
type algorithm called auto-conditioned fast gradient method (AC-FGM) that can
achieve an optimal $\mathcal{O}(1/k^2)$ rate of convergence for smooth convex
optimization without requiring the estimate of a global Lipschitz constant or
the employment of line search procedures. We then extend AC-FGM to solve convex
optimization problems with H\"{o}lder continuous gradients and show that it
automatically achieves the optimal rates of convergence uniformly for all
problem classes with the desired accuracy of the solution as the only input.
Finally, we report some encouraging numerical results that demonstrate the
advantages of AC-FGM over the previously developed parameter-free methods for
convex optimization.
