Abstract
We study necessary conditions for an optimization problem with the objective and constraint functions including extreme functions. The extreme function is a function which is defined by max or min operations with respect to the opponent's variable, and hence it is a non-differentiable function in general. So, we are concerned with the optimization problems with non-differentiable objective and constraint functions.
We present a new approach to the necessary conditions for non-differentiable optimization problems by means of generalized Farkas' lemma, and also, as an application, deal with optimality conditions for the optimization problem mentioned above.
At the start, the necessary conditions are described by a system of inequalities with general directional derivatives of objective and constraint functions. Then we apply the generalized Farkas' lemma to the system of inequalities to obtain the Kuhn-Tucker type conditions. We also present the chain-rule of directional derivatives.
