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.

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 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.

Hypergroups that have at least one identity element and where each element has at least one inverse are called Regular hypergroup. In this regards, for a Regular hypergroup $H$, it is shown that there exists a correspondence between the set of all strongly Regular relations on $H$ and the set of all normal subhypergroups of $H$ containing $S_{\beta}$. More precisely, it has been proven that for every strongly Regular relation $\rho$ on $H$, there exists a unique normal subhypergroup of $H$ containing $S_{\beta}$, such that its quotient is a group, isomorphic to $H/\rho$. Furthermore, this correspondence is extended to a lattice isomorphism between them.

In this paper we study a discrete time version of deterministic models in optimization in infinite dimensional. The functional are assumed to be merely lower semi continuous. We obtain optimality conditions which are always necessary and which are also sufficient in the convex case whenever the given problem satisfies a qualification condition.

This paper addresses the problem of fixed-time control for a class of second-order nonlinear systems in the presence of model uncertainty and external disturbance. By introducing a novel form of non-singular terminal sliding mode control, a fixed-time control is designed to obtain acceptable performance, rapid convergence of the system states, high robustness and singularity elimination. Guaranteeing fixed-time convergence is a significant feature of the proposed control law under which the convergence time of the proposed surface is independent of the initial conditions. Since the upper bound of the system uncertainty and disturbance is quite difficult to obtain, an adaptive mechanism is presented under which there is no need to know this upper bound. Lyapunov analysis proves that the system states converge to small neighborhood of the origin within a fixed time. To assess efficiency of the suggested method, a flexible spacecraft attitude control system is considered and a fixed-time attitude control system is derived. Simulation results verify the effectiveness and performance of the presented approach.