Adaptive Accelerated Gradient Method for Smooth Convex Optimization

Abstract We propose an adaptive accelerated gradient method for solving smooth convex optimization problems. The method incorporates a scheme to determine the step size adaptively, by means of a local estimation of the smoothness constant, which is assumed unknown, without resorting to line search procedures. The sequence generated by this method converges weakly to a minimizer of the objective function, and the function values converge at a fast rate of $$\mathcal {O}\left( \frac{1}{k^2} \right) $$ O 1 k 2 . Moreover, if the objective function is strongly convex, the function values converge at a linear rate $$\mathcal {O}\left( \frac{1}{k^2}(1-\rho )^k \right) $$ O 1 k 2 ( 1 - ρ ) k , with $$\rho =\mathcal {O}\left( \frac{\mu }{L} \right) $$ ρ = O μ L , without knowledge of the strong convexity parameter.