Derivatives Of Set-valued Maps Research Articles

Second-Order Composed Radial Derivatives of the Benson Proper Perturbation Map for Parametric Multi-Objective Optimization Problems

In this paper, we introduce second-order composed radial derivatives of set-valued maps and establish some of its properties. By applying this second-order derivative, we obtain second-order sensitivity results for parametric multi-objective optimization problems under the Benson proper efficiency without assumptions of cone-convexity and Lipschitz continuity. Some of our results improve and derive the recent corresponding ones in the literature.

Second-order optimality conditions for set optimization using coradiant sets

The aim of this paper is to study second-order optimality conditions for a set-valued optimization problem with set criterion, where the order relation is induced by a set belong to a class of specific coradiant sets and is not necessarily a preorder. We introduce a notion of generalized second-order radial set, different from the classical second-order radial set, it is defined on a set, not on a point. Using the generalized second-order radial set, we introduce a new type of generalized second-order radial derivatives for set-valued maps and apply them to establish some necessary and sufficient conditions for the lower strict minimal solution of the set optimization problem.

The second-order contingent derivative of generalized perturbation maps

In the paper, we give some remarks on [1]. Then, we modify main results concerning the sum rule of second-order contingent derivatives for set-valued maps and its application to the sensitivity analysis of generalized perturbation maps. The obtained results are new and better than those in [1]. Some examples are proposed to illustrate our results.

Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints

In this paper, we propose the concept of second-order composed adjacent contingent derivatives for set-valued maps and hence discuss the included relationship to the second-order composed contingent derivatives. Under the appropriate conditions, Lipschitz properties of the first order derivatives and a chain rule for such second-order composed contingent derivatives are demonstrated. By virtue of second-order composed adjacent contingent derivatives and second-order composed contingent derivatives, we establish second-order Karush–Kuhn–Tucker sufficient and necessary optimality conditions for a set-valued optimization problem subject to mixed constraints. Applying a separation theorem for convex sets and cone-Aubin properties, we address stronger necessary optimality conditions and extend the second-order Kurcyusz–Robinson–Zowe regularity assumption for a problem with mixed constraints, in which the derivatives of the objective and the constraint functions are considered in separated ways. When the results regress to vector optimization, we also extend and improve some recent existing results.

Derivatives of Set-Valued Maps and Gap Functions for Vector Equilibrium Problems

This paper deals with the set-valued gap functions for vector equilibrium problems and investigates their differential properties using Hadamard directional differentials. Also, contingent and adjacent derivatives of a class of set-valued maps are characterized. Moreover, some basic properties of Φ-contingent and Φ-adjacent cones are given in the presence of a nonsmooth kernel function.

Morphological Transition of Schooling in 19th-Century France

The transformation over time of level sets—the sets of states for which a given function takes a given value—has no reason to be regular, especially in social science, where the structure of these level sets of a given variable may be altered by a fluctuating demography or disrupted by unexpected events, and where the actors contribute to shaping their space through investment policies. The pioneering construction of graphic derivatives of set-valued maps renews the understanding of dynamic level sets in social science. The exemplary case of schooling in 19th-century France shows the set derivatives of level sets and illustrates how the Franco-Prussian War of 1870–1871 disturbed the directions taken in 1867. A comparison of the observed changes of level sets with those observed under rational theoretical policies leads to the conclusion that the actors behaved with the ambition of optimizing the overall schooling incidence while reducing the gap between boys and girls.

Characterizing Generalized Derivatives of Set-Valued Maps: Extending the Tangential and Normal Approaches

For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [C. H. J. Pang, Math. Oper. Res., 36 (2011), pp. 377--397]. This result generalizes the Aubin criterion in [A. L. Dontchev, M. Quincampoix, and N. Zlateva, J. Convex Anal., 3 (2006), pp. 45--63]. A second characterization of these generalized derivatives is easier to check in practice, especially in the finite dimensional case. Finally, the third characterization in terms of limiting normal cones and coderivatives generalizes the Mordukhovich criterion in the finite dimensional case. The convexified coderivative has a bijective relationship with the set of possible generalized derivatives. We conclude by illustrating a few applications of our result.

Continuity of second-order contingent derivative of the efficient set map for parametrized multiobjective optimization

In this paper, some relationships between the second-order contingent derivative of a set-valued map and its profile map are obtained. By virtue of the second-order contingent derivatives of set-valued maps, upper semicontinuity and lower semicontinuity are obtained for efficient set maps of parametrized multiobjective optimization. Several examples are provided to show the results obtained.

Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions

In this paper we give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second-order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. An application to a special type of vector optimization problems, where the objective is given as the sum of two multifunctions, is presented. Furthermore, also as application, a special attention is paid for the case of perturbation set-valued maps which naturally appear in optimization problems.

Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization

In this paper, generalized higher-order contingent (adjacent) derivatives of set-valued maps are introduced and some of their properties are discussed. Under no any convexity assumptions, necessary and sufficient optimality conditions are obtained for weakly efficient solutions of set-valued optimization problems by employing the generalized higher-order derivatives.

Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

By virtue of higher-order adjacent derivative of set-valued maps, relationships between higher-order adjacent derivative of a set-valued map and its profile map are discussed. Some results concerning stability analysis are obtained in parametrized vector optimization.

A Set-Valued Analysis Approach to Second Order Differentiation of Nonsmooth Functions

A theory of second order differentiability for real functions with calm Clarke gradient at points of strict differentiability is presented from a set-valued analysis perspective by using the theory of graphical derivatives of set-valued maps. Connections with other second order theories and applications in optimization are given.

Epidifferentiability of the map of infima in Hilbert spaces

This article deals with derivatives for set-valued maps that take values in ordered vector spaces, in particular it concerns about the relationship between the epiderivatives of a set-valued map and its associated map of infima. When the image space is a real separable Hilbert space ordered by an orthonormal basis, by using a variational technique based on a decoupling of the ordering cone into half-spaces, we show that both epiderivatives coincide under certain hypothesis of compactness and stability. Furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set-valued maps.

Variational inclusions for contingent derivative of set-valued maps

In this paper, we give two versions of Ky Fan's inequality for set-valued maps acting between normed vector spaces and we consider sufficient conditions to solve a variational inclusion problem concerning derivatives of set-valued maps. A selection result for set-valued maps between finite dimensional vector spaces and its contingent derivative is obtained as well; from this result we derive some conditions for the existence of a solution of a generalized variational inequality problem.