NONSMOOTH INVEX FUNCTIONS VIA UPPER DIRECTIONAL DERIVATIVE OF DINI

The notion of invexity is introduced by Hanson [6] ˆın 1981, for real differentiable functions, see also [2] by Craven. After the works of Hanson and Craven, other types of differentiable functions have appeared with the intent of generalizing invex functions from different viewpoints, see [7] by Jeyakumar, [8] by Kaul and Kaul and [15], [16], [22] by Mititelu. Stipulating possible applications of these functions for programming problems, see [11] by Mishra and [24] by Nahak and Mohapatra to quote some examples, some scientists payed attention to possible extensions of this concept to nonsmooth functions. A direction is those of Craven [3] followed by Reiland [26], which extended the concept of invexity to nonsmooth functions by using the generalized directional derivative of Clarke for Lipschitz functions. Another direction is those of Mititelu, which extended the concept of invexity to nonsmooth functions by using essentially the upper directional derivative of Dini [5], [13], [16], [17], [18], [19], [22]. The goal of this work is to describe the state of the art on the question of invex functions for this case. We recall the main results on this theory with proofs when necessary, we supply up to date references, a description of some further developments and a few new results as well. For additional background material on this topic, we address the reader to the following research works: [10], [12], [14], [21], [23]