On the direction of the steepest descent

The article is dedicated of memory of Professor V. F. Demyanov (1938-2014). The main scientific interests of V. F. Demyanov lay in the field of numerical optimization methods, where the notion of the direction of the steepest descent plays an important role. This notion is introduced for both smooth and nonsmooth functions, both in constrained and unconstrained cases. This paper provides a detailed analysis of methods for constructing the direction of steepest descent. In all cases, it comes down to solving the quadratic programming problem. Particular attention is paid to nonsmooth functions, in the study of which V. F. Demyanov made a significant contribution. An example of a function which is quasidifferentiable at a point is given. This function has two directions of the steepest descent and two directions of the steepest ascent.