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