### Abstract:

In spite of the fact that the origin of non-smooth optimization as a mathematical discipline is quite
recent; it is now well established as an important and very active branch of applied mathematics. Many
practical problems in economics, physics, aerospace, as well as other areas of applications cannot be
adequately described without the help of non-smooth functions.
In the theory of optimization several types of piecewise differentiable functions occur in quite natural
way. As a typical example for such non-differentiable functions we mention the finite max-min
combinations of differentiable functions. A more general class is the quasidifferentiable functions
which are investigated in detail by V.F.Demyanov and A.M.Rubinov. The directional derivatives of
these functions can be represented as a difference of two sublinear functions. Since a sublinear
function is uniquely described by its subdifferential in the origin, there exists a natural correspondence
between the directional derivatives and the set of pairs of compact convex sets. However, this
representation is not unique. This nonuniqueness inspires mathematicians to find a minimal
representation of the directional derivative, which is equivalent to finding a minimal pair of compact
convex sets. In this project, the theory of quasidifferentiable optimization and minimal pairs of
compact convex sets is discussed. In the first chapter, general introduction and description of the
problem are given. In the second chapter basic definitions and concepts are mentioned and in the last
chapter the detail discussion of minimal pairs of compact convex sets is given.