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 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.

In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are locally Lipschitz.Necessary optimality conditions and regularity conditions are given. Our approach are based on the Michel-Penot subdifferential.

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.

This article is concerned with non-smooth/nonconvex robust multi-objective optimization problems involving uncertain inequality and equality constraints. Employing some advanced tools of variational analysis such as the approximate extremal principle and the weak fuzzy sum rule for the Frechet subdifferential, we first drive a fuzzy necessary optimality condition of a non-smooth/nonconvex multi-objective optimization problem without any constrained qualification in the sense of the Frechet subdifferential. Then by exploiting the obtained fuzzy optimality condition, the non-smooth version of Fermat’, s rule and formulae for the limiting subdifferential of an infinite family of non-smooth functions, we establish a necessary optimality condition in terms of the limiting subdifferential for weakly robust efficient solutions of the reference problem. Further, we present an example to illustrate this condition for an uncertain multi-objective optimization problem involving equality and inequality constraints. Finally sufficient conditions for weakly robust efficient solutions and robust efficient solutions of the problems are provided by presenting new concepts of generalized convexity.

This survey investigates some developments in the second-order characterization of generalized convex functions using the coderivative of subdifferential mapping. More precisely, it presents the second-order characterization for quasiconvex, pseudoconvex and invex functions. Furthermore, it gives some applications of the second-order subdifferentials in optimization problems such as constrained and unconstrained nonlinear programming.

In this paper, some properties of  pseudoinvex functions, defined by means of  limiting subdifferential, are discussed. Furthermore, the Minty vector variational-like inequality,  the Stampacchia vector variational-like inequality, and the  weak formulations of these two inequalities  defined by means of limiting subdifferential are studied. Moreover, some relationships  between the vector variational-like inequalities and vector optimization problems are established.

We introduce some new concepts of locally Lipschitz mappings, Clarke subdifferential, approximate convexity and sub-monotonocity in locally convex spaces. We show that, if f is approximately convex and bounded above, then f is locally Lipschitz. We also prove that a Lipschitz function is approximately convex if and only if its Clarke subdifferential is a submonotone operator. Several properties of approximate convexity are discussed. Our results can be viewed as extensions and refinements of the previously known results from Banach spaces to locally convex spaces. 

In this paper, multiobjective optimization problems with nondifferentiable quasiconvex functions are considered. We obtain some duality results and a linear representation for the considered problems. Since the well-known strong duality result is not valid for the problems, we present a weaker form of it, named quasi-strong duality result.