We are interested in proposing the convex fuzzy functions from R into E to describe the concepts of left and right Generalized derivatives. In this way, several properties of the concepts are discussed. As well, the notions of Generalized subdifferential for these functions are studied and calculated. Finally, the properties of the concepts are illustrated and applied through some examples.

In this paper, we deal with the subdifferential concept on Hadamard spaces. Flat Hadamard spaces are characterized, and necessary and sufficient conditions are presented to prove that the subdifferential set in Hadamard spaces is nonempty. Proximal subdifferential in Hadamard spaces is addressed and some basic properties are highlighted. Finally, a density theorem for subdifferential set is established.

‎In this paper‎, ‎we extend the notion of approximate convexity to set-valued maps and obtain some relations between approximate convexity and approximate monotonicity of their normal subdifferential‎.

In this paper, we investigate relation between weak subdifferential and augmented normal cone. We define augmented normal cone via weak subdifferential and vice versa. The necessary conditions for the global maximum are also stated. We produce preliminary properties of augmented normal cones and discuss them via the distance function. Then we obtain the augmented normal cone for the indicator function. Relation between weak subifferential and augmented normal cone and epigraph is also explored. We also obtain optimality conditions via weak subdifferential and augmented normal cone. Finally, we define the Stampacchia and Minty solution via weak subdifferential and investigate the relation between Stampacchia and Minty solution and the minimal point.

The purpose of this paper is to develop nonsmooth optimization problems (P) in which all emerging functions are assumed to be real-valued quasiconvex functions that are defined on a finite-dimensional Euclidean space. First, we introduce two linear optimization problems with the same optimal value of the considered problem. Then, we introduce a real-valued non-negative gap function for (P), and we provide some conditions which ensure that its null points are the same as the optimal solution of problem (P). The results are based on incident subdifferential, which is an important concept in the analysis of quasiconvex functions.

This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective Generalized semi-infinite programming problems‎. ‎These problems involve inequality constraints whose index set depends on the decision vector‎, ‎and all emerging functions are assumed to be locally Lipschitz‎. ‎We introduce a new constraint qualification for these problems‎. ‎Building upon this qualification‎, ‎we derive an upper estimate for the Clarke sub-differential of the value function of the problem‎. ‎Furthermore‎, ‎we demonstrate the necessary optimality conditions for properly efficient solutions to the problem‎.