Quasidifferentiable Optimization and Minimal Pairs of Compact Convex Sets

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Addis Ababa University


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.



Faculty of Computer and Mathematical