Optimality and algorithm for generalized invex Chebyshev problem

In this paper Clarke's generalized directional derivative is used to establish necessary and sufficient optimality conditions for the Chebyshev problem with mixed constraints (inequality and equality constraints). For sufficient conditions it is furthermore assumed that the functions which define the object of the problem are pseudoinvex, those which define the inequality constraints are quasiinvex and those which define the equality constraints are inquasimonotonic. Moreover, if the functions of the problem are differentiable, we establish a new criterion of sufficient optimality of an admissible solution. Using this criterion, when the functions of the problem are also ‘strict’, this paper presents an algorithm of the admissible directions type to solve Chebyshev's problem. A numerical example is given as an illustration.