Radial Epiderivatives Research Articles

Generalized Derivatives and Optimality Conditions in Nonconvex Optimization

In this paper, we study the radial epiderivative notion for nonconvex functions, which extends the (classical) directional derivative concept. The paper presents new definition and new properties for this notion and establishes relationships between the radial epiderivative, the Clarke’s directional derivative, the Rockafellar’s subderivative and the directional derivative. The radial epiderivative notion is used to establish new regularity conditions without convexity conditions. The paper presents explicit formulations for computing the radial epiderivatives in terms of weak subgradients and vice versa. We also present an iterative algorithm for approximate computing of radial epiderivatives and show that the algorithm terminates in a finite number of iterations. The paper analyzes necessary and sufficient conditions for global optimums in nonconvex optimization via the radial epiderivatives. We formulate a necessary and sufficient condition for a global descent direction for radially epidifferentiable nonconvex functions. All the properties and theorems presented in this paper are illustrated and interpreted on examples.

Optimality Conditions for Approximate Solutions of Set Optimization Problems with the Minkowski Difference

In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained.

New second-order radial epiderivatives and applications to optimality conditions

In this paper, we introduce the second-order weakly composed radial epiderivative of set-valued maps, discuss its relationship to the second-order weakly composed contingent epiderivative, and obtain some of its properties. Then we establish the necessary optimality conditions and sufficient optimality conditions of Benson proper efficient solutions of constrained set-valued optimization problems by means of the second-order epiderivative. Some of our results improve and imply the corresponding ones in recent literature.

An extremal vector problem under inclusion constraints via radial epi-derivatives

An extremal vector problem under inclusion constraints via radial epi-derivatives

Optimality Conditions for Nonconvex Nonsmooth Optimization via Global Derivatives

The notions of upper and lower global directional derivatives are introduced for dealing with nonconvex and nonsmooth optimization problems. We provide calculus rules and monotonicity properties for these notions. As a consequence, new formulas for the Dini directional derivatives, radial epiderivatives and generalized asymptotic functions are given in terms of the upper and lower global directional derivatives. Furthermore, a mean value theorem, which extend the well-known Diewert’s mean value theorem for radially upper and lower semicontinuous functions, is established. We also provide necessary and sufficient optimality conditions for a point to be a local and/or global solution for the nonconvex minimization problem. Finally, applications for nonconvex and nonsmooth mathematical programming problems are also presented.

New second-order radial epiderivatives and nonconvex set-valued optimization problems

In this paper, we introduce the generalized second-order composed radial epiderivative of set-valued maps and establish a few relations between the epiderivative and the generalized second-order composed contingent epiderivative. We also investigate some of its properties. Then, by virtue of the generalized second-order composed radial epiderivative, we obtain the necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions of unconstrained set-valued optimization problems without convexity conditions. Furthermore, the necessary optimality conditions and sufficient optimality conditions are obtained for Henig efficient solutions of constrained set-valued optimization problems. Some of obtained results improve or imply the corresponding ones in recent literature.

Generalized composed radial epiderivatives

Generalized composed radial epiderivatives

Higher-order weak radial epiderivatives and non-convex set-valued optimization problems

<p style='text-indent:20px;'>In the paper, we propose the notion of the higher-order weak radial epiderivative of a set-valued map, and discuss some of its properties. Then, by virtue of the higher-order weak radial epiderivative, we establish the optimality necessary conditions and sufficient ones of weak efficient solutions (Pareto efficient solutions) for non-convex set-valued optimization problems. Some of the obtained results improve and extend the recent existing results. Several examples are provided to show the main results obtained.

Higher-Order Generalized Radial Epiderivative and Its Applications to Set-Valued Optimization Problems

In the paper, we introduce the higher-order generalized radial epiderivative of set-valued maps. Then, its basic properties are studied. Finally, we apply this epiderivative to optimality conditions for set-valued optimization problems.

Existence theorems for second-order radial epiderivatives

Existence theorems for second-order radial epiderivatives

Optimality conditions of second-order radial epiderivatives

In this paper, we introduce the concepts of second-order radial epiderivative and second-order generalized radial epiderivative for nonconvex set-valued maps. We give existence theorems for the second-order generalized radial epiderivatives. We also establish the second-order optimality conditions by using second-order radial epiderivatives. https://doi.org/10.28919/jmcs/3440

Mixed Type Duality for Set-valued Optimization Problems via Higher-order Radial Epiderivatives

ABSTRACTIn this article, we introduce a notion of higher-order radial epiderivative for set-valued maps and study its properties. A generalized concept of higher-order strict minimizers in set-valued optimization is proposed as well. By virtue of the radial epiderivative, we establish a mixed dual problem, and then weak, strong, and converse duality theorems are obtained in dealing with generalized strict minimizers.

Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives

This paper deals with maximization and minimization of quasiconvex functions in a finite dimensional setting. Firstly, some existence results on closed convex sets, possibly containing lines, are presented. This is given via a careful study of reduction to the boundary and/or extremality of the feasible set. Necessary or sufficient optimality conditions are derived in terms of radial epiderivatives. Then, the problem of minimizing quasiconvex functions are analyzed via asymptotic analysis. Finally, some attempts to define asymptotic functions under quasiconvexity are also outlined. Several examples illustrating the applicability of our results are shown.

On generalized $$\epsilon $$ ϵ -subdifferential and radial epiderivative of set-valued mappings

In this paper, a generalized \(\epsilon \)-subdifferential, which was defined by the radial epiderivative and a norm, is first introduced for a set-valued mapping. Some existence theorems of the generalized \(\epsilon \)-subdifferential and the radial epiderivative are discussed. A relationship between the existence of the radial epiderivative and the existence of the generalized \(\epsilon \)-subdifferential is investigated for a set-valued mapping.

Extended radial epiderivatives of non-convex vector-valued maps and parametric quasiconvex programming

In this paper, we consider extended vector-valued mappings defined on a normed linear space. Based on the recent semicontinuous regularizations related to hypographical and/or epigraphical profile mappings of the considered function introduced, we define semicontinuous radial epiderivatives. We, then, demonstrate that the properties of these epiderivatives amount to properties of hypographical and/or epigraphical profile mappings of the corresponding difference quotient of the underlying function, which simplify fairly well the proofs in the radial epiderivative formulaes. In particular, we stress the impact of semicontinuity, hence, we characterize with new arguments the radial epiderivatives in terms of the suprema and/or infima of the interiorly radial cone of the hypograph and/or epigraph of the considered function. Finally, we obtain optimality conditions for general non-convex constrained vector optimization problems. We apply thereafter the obtained pattern to a parametric quasiconvex programming problem for which we derive necessary and sufficient optimality conditions that are not sensitive to perturbation at the nominal level, yielding henceforth more – and strong at least under asymptotically regular constraints – information than the recent stability results obtained under additional conditions on the regularity of the normal cone to the adjusted sublevel sets of the underlying function.

Some relationships among gw-subdifferential, directional derivative and radial epiderivative for nonconvex vector functions

In this work, we give some characterizations of gw-subdifferentiability of a vector-valued function by using its directional derivative and radial epiderivative. Moreover, under some assumptions, we proved that the directional derivative and radial epiderivative of a vector-valued function are the elements of the supremum set of gw-subgradients of it. Finally, without any convexity assumption, we proved that the epigraph of contingent derivative of a set valued map is included in the epigraph of contingent epiderivative of this set-valued map.

Generalized radial epiderivatives and nonconvex set-valued optimization problems

In this article, the generalized radial epiderivative for a set-valued map is introduced. An existence result of generalized radial epiderivatives is obtained. Moreover, the relation between these kind of derivatives and radial epiderivatives has been established. Based on the generalized radial epiderivative, necessary and sufficient optimality conditions are derived for nonconvex set-valued optimization problems.

Optimality conditions for vector equilibrium problems in normed spaces

By using the concepts of contingent epiderivative, radial epiderivative, Clarke tangent epiderivative and Y-epiderivative, we present necessary and sufficient conditions for the weakly efficient solution, the Henig efficient solution, and the globally proper efficient solution, respectively, to vector equilibrium problems with constraints.

$\mathbb{T}$-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities

In this paper, we first define a $\mathbb{T}$-cone which is a unified version of several cones, namely, contingent cone, radial cone, $C$-tangent cone, Clarke tangent cone, $S$-cone, adjacent cone, etc. Then, we define the $\mathbb{T}$-epiderivative of a set-valued map which includes the contingent epiderivative, radial epiderivative, $S$-epiderivative, adjacent epiderivative etc. as special cases. We present several properties of such an epiderivative. The generalized vector $\mathbb{T}$-variational inequality problem is also considered. We provide necessary and sufficient conditions for a solution of a set optimization problem. Several existence results for solutions of set optimization problems and a generalized vector $\mathbb{T}$-variational inequality problem are given.

T-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities

In this paper, we first define a T-cone which is a unified version of several cones, namely, contingent cone, radial cone, C-tangent cone, Clarke tangent cone, S-cone, adjacent cone, etc. Then, we define the T-epiderivative of a set-valued map which includes the contingent epiderivative, radial epiderivative, S-epiderivative, adjacent epiderivative etc. as special cases. We present several properties of such an epiderivative. The generalized vector T-variational inequality problem is also considered. We provide necessary and sufficient conditions for a solution of a set optimization problem. Several existence results for solutions of set optimization problems and a generalized vector T-variational inequality problem are given.