An extension of Hestenes necessary condition

After the pioneering approach of Hestenes, which has not been sufficiently considered, the concept of image of a constrained extremum problem has been recognized recently to be a powerful tool in optimization. Hence in the is paper, we consider a necessary optimality condition for a nondifferentiable extremum problem established by Hestenes in the image space and, by again exploiting the concept of image of the problem, a necessary condition embodying that of Hestenes is given. Moreover, the study is completed by expressing the previous result in the space where the unknowns of the given problem run. In this way, we obtain a Lagrangian-type necessary condition holding for a particular class of semidifferentiable problems.