  It is well-known that accelerated gradient first order methods possess
optimal complexity estimates for the class of convex smooth minimization
problems. In many practical situations, it makes sense to work with inexact
gradients. However, this can lead to the accumulation of corresponding
inexactness in the theoretical estimates of the rate of convergence. We propose
some modification of the methods for convex optimization with inexact gradient
based on the subspace optimization such as Nemirovski's Conjugate Gradients and
Sequential Subspace Optimization. We research the method convergence for
different condition of inexactness both in gradient value and accuracy of
subspace optimization problems. Besides this, we investigate generalization of
this result to the class of quasar-convex (weakly-quasi-convex) functions.
