  The work is devoted to the construction of efficient and applicable to real
tasks first-order methods of convex optimization, that is, using only values of
the target function and its derivatives. Construction uses OGM-G, fast gradient
method which is optimal by complexity, but requires to know the Lipschitz
constant for gradient and the strong convexity constant to determine the number
of steps and step length. This requirement makes practical usage impossible. An
adaptive on the constant for strong convexity algorithm ACGM is proposed, based
on restarts of the OGM-G with update of the strong convexity constant estimate,
and an adaptive on the Lipschitz constant for gradient ALGM, in which the use
of OGM-G restarts is supplemented by the selection of the Lipschitz constant
with verification of the convexity conditions used in the universal gradient
descent method. This eliminates the disadvantages of the original method
associated with the need to know these constants, which makes practical usage
possible. Optimality of estimates for the complexity of the constructed
algorithms is proved. To verify the results obtained, experiments on model
functions and real tasks from machine learning are carried out.

Optimization and Control