Quasidifferentiability and quasidifferential optimization

The notion of quasidifferential was introduced by V.F.Demyanov and A.M.Rubinov thirty years ago. Quasidifferentiable functions has a great role since all directionally differentiable Lipschitzian functions can be approximated by quasidifferentiable functions.In this work, which is consisted of four chapters, an introduction to functions which can be expressed as the difference of two convex positively homogeneous functions is made; quasidifferentiable functions are defined and the relationship between quasidifferentials and the pairs of compact convex sets is studied. Also, necessary and sufficient conditions for a solution of extremum problems quasidifferentiablefunction are given.In the first chapter of this work, some basic definitions and theorems , necessary for this work, are given. In the second chapter, the notion of DC function is defined and its some property is investigated. After that, DC representation of some special functions is investigated and an introduction to DC sets is made.In the third chapter,by defining the quasidifferential some properties of quasidifferentiable functions is invastigated. Also, the relationship between quasidifferentialsand the pairs of compact convex sets is given and quasidifferential of some special functions is expressed with the pairs of compact convex sets. After that, the quasidifferentiability of operators, which are de¯ned on Banach spaces, is defined.In the last chapter, necessary and sufficient conditions for a solution of extremum problems, which has quasidifferentiable nonconvex objective function, are given.