Conjugacy, quasidifferentiability and nonconvex optimization

Weak lower/upper exhausters of positively homogeneous functions defined by Küçük et al. were constructed via relationships between weak subdifferential/superdifferential and exhausters, and they presented a special class of exhausters. In the same study, some geometric methods to calculate weak exhausters were given by using Minkowski sum and difference operations. Some necessary and sufficient optimality conditions were also given via weak exhausters. In addition, it was shown that weak lower (upper) exhausters can be reduced by indexing them only with the boundary points of weak subdifferential (weak superdifferential). Some optimality conditions were also expressed via reduced weak exhausters. In this study, relationships between (weak) conjugate function and (weak) exhausters are investigated. Firstly, weak subdifferential of a function is expressed by means of the weak conjugate of this function and this new expression provides a characterization of weak lower exhausters in terms of weak conjugate function. Some new optimality conditions are obtained for maximization problems by using this characterization. After that, a new optimization problem is constructed which admits the directional derivative of the objective function of a convex optimization problem as the objective function. Solutions of the conjugate dual of this problem are given by means of the sets belonging to any upper exhauster of the value function of the primal problem. Additionally, considering nonconvex optimization problems, solutions of weak Fenchel dual problems of primal problems are represented by weak upper exhausters. In this way, some methods are given to determine stationary points of the primal problems for both cases. Minimization of quasidifferentiable functions are also investigated and a DC programming problem is constructed which admits the dc-directional derivative of the quasidifferentiable objective function of an optimization problem as the objective function. Some characterizations are obtained to find solutions of this problem and the DC dual of it. In addition, a characterization for solutions of weak Fenchel duals of this type of problems are also given.