The generalized Myshkis problem consists in analysis of robust stability of time-delay system under variation of its parameters within a set in an infinite-dimensional space. In our paper, we consider a linear system, where the delay is a bounded function under an additional restriction on its growth rate. Some sufficient conditions of solvability of the generalized Myshkis problem are expressed in the form of quadratic convex optimization task. The new approach is based on a modification of the Lyapunov-Razumikhin method.