Second-Order Differential and Sensitivity Properties of Weak Vector Variational Inequalities

In this paper, by using the second-order contingent derivative, second-order differential properties of a class of set-valued maps are investigated and an explicit expression of the second-order contingent derivatives is obtained. Then, by means of a gap function, second-order sensitivity properties are discussed for a weak vector variational inequality.     

Second-order asymptotic differential properties and optimality conditions for weak vector variational inequalities

In this paper, by virtue of an asymptotic second-order contingent derivative and an asymptotic second-order Φ-contingent cone, differential properties of a class of set-valued maps are investigated and an explicit expression of their asymptotic second-order contingent derivatives is established. Then, second-order necessary optimality conditions of solutions are obtained for weak vector variational inequalities.     

New second-order contingent epiderivatives and set-valued optimization problems

In this paper, we introduce the concept of second-order compound contingent epiderivative for set-valued maps and discuss its relationship to the second-order contingent epiderivative. Simultaneously, we also investigate some special properties of the second-order compound contingent epiderivative. By virtue of the second-order compound contingent epiderivative, we establish some unified second-order sufficient and necessary optimality conditions for set-valued optimization problems. All results in this paper generalize the corresponding results in the literature.     

New Generalized Second-Order Contingent Epiderivatives and Set-Valued Optimization Problems

In this paper, we introduce the concept of a generalized second-order composed contingent epiderivative for set-valued maps and discuss its relationship to the generalized second-order contingent epiderivative. We also investigate some of its properties. Then, by virtue of the generalized second-order composed contingent epiderivative, we establish a unified second-order sufficient and necessary optimality condition for set-valued optimization problems, which is a generalization of the corresponding results in the literature.     

集值向量优化的二阶最优性必要条件

在赋范空间中给出了集值映射的二阶切集的概念,利用二阶切集,定义了集值映射的二阶切导数。然后,获得了集值向量优化问题弱极小元的两个二阶最优性必要条件。     

Second-Order Optimality Conditions for Strict Efficiency of Constrained Set-Valued Optimization

In this paper, we propose several second-order derivatives for set-valued maps and discuss their properties. By using these derivatives, we obtain second-order necessary optimality conditions for strict efficiency of a set-valued optimization problem with inclusion constraints in real normed spaces. We also establish second-order sufficient optimality conditions for strict efficiency of the set-valued optimization problem in finite-dimensional normed spaces. As applications, we investigate second-order sufficient and necessary optimality conditions for a strict local efficient solution of order two of a nonsmooth vector optimization problem with an abstract set and a functional constraint.     

Sensitivity and stability for the second-order contingent derivative of the proper perturbation map in vector optimization

Sensitivity analysis and stability analysis in vector optimization are dealt with in this paper. First, some relationships between the second-order contingent derivative of a set-valued map and its profile map are obtained. Secondly, the upper semicontinuity and lower semicontinuity of second-order contingent derivatives of set-valued maps are established. Finally, by virtue of the second-order contingent derivative of set-valued maps, quantitative information and qualitative information on the behavior of the proper perturbation map are obtained.     

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.     

Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization

In this paper, we propose the concept of a second-order composed contingent derivative for set-valued maps, discuss its relationship to the second-order contingent derivative and investigate some of its special properties. By virtue of the second-order composed contingent derivative, we extend the well-known Lagrange multiplier rule and the Kurcyusz–Robinson–Zowe regularity assumption to a constrained set-valued optimization problem in the second-order case. Simultaneously, we also establish some second-order Karush–Kuhn–Tucker necessary and sufficient optimality conditions for a set-valued optimization problem, whose feasible set is determined by a set-valued map, under a generalized second-order Kurcyusz–Robinson–Zowe regularity assumption.     

Generalized second-order contingent epiderivatives in parametric vector optimization problems

This paper is concerned with generalized second-order contingent epiderivatives of frontier and solution maps in parametric vector optimization problems. Under some mild conditions, we obtain some formulas for computing generalized second-order contingent epiderivatives of frontier and solution maps, respectively. We also give some examples to illustrate the corresponding results.     

A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

A variational inequality theory for demicontinuous S-contractive maps in Hilbert spaces is established by employing the ideas of Granas' topological transversality. Such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward S-contractive maps and to those of fixed point index for condensing maps. The variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second-order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical Sobolev exponent. There has been little discussion for such elliptic inequalities involving the critical Sobolev exponent in the literature.     

Second-Order Optimality Conditions in Set Optimization

In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.The authors are grateful to the referees for careful reading and helpful remarks.     

变序结构局部弱非控点的二阶刻画

引进了一种二阶切导数,借助该切导数给出了变序结构集值优化问题取得局部弱非控点的二阶最优性必要条件.在某种特殊情况下,给出了一阶最优性条件.通过修正的Dubovitskij-Miljutin切锥导出的约束规格,给出了两个集值映射之和的二阶相依切导数的关系式,进一步得到目标函数与变锥函数的二阶相依切导数分开形式的最优性必要条件.     

Controllability of second-order neutral functional differential inclusions in Banach spaces

In this paper, we prove the controllability of second-order neutral functional differential inclusions in Banach spaces. The result are obtained by using the theory of strongly continuous cosine families and a fixed point theorem for condensing maps due to Martelli.     

锥上减映象的不动点及其在非局部边值问题中的应用

研究锥上一类减映象,得到这类映象不动点的存在性、唯一性和迭代序列的收敛性.其结果在二阶非局部边值问题中得到应用.     

On Quantitative Stability in Optimization and Optimal Control

We study two continuity concepts for set-valued maps that play central roles in quantitative stability analysis of optimization problems: Aubin continuity and Lipschitzian localization. We show that various inverse function theorems involving these concepts can be deduced from a single general result on existence of solutions to an inclusion in metric spaces. As applications, we analyze the stability with respect to canonical perturbations of a mathematical program in a Hilbert space and an optimal control problem with inequality control constraints. For stationary points of these problems, Aubin continuity and Lipschitzian localization coincide; moreover, both properties are equivalent to surjectivity of the map of the gradients of the active constraints combined with a strong second-order sufficient optimality condition.     

A stable semi-discrete central scheme for the two-dimensional incompressible Euler equations

^** Email: dlevy{at}math.stanford.edu We derive a second-order, semi-discrete central-upwind schemefor the incompressible 2D Euler equations in the vorticity formulation.The reconstructed velocity field preserves an exact discreteincompressibility relation. We state a local maximum principlefor a fully discrete version of the scheme and prove it usinga convexity argument. We then show how similar convexity argumentscan be used to prove that the scheme maps certain Orlicz spacesinto themselves. The consequences of this result on the convergenceof the scheme are discussed. Numerical simulations support theexpected properties of the scheme.     

Metric regularity,tangent sets,and second-order optimality conditions

A strong regularity theorem is proved, which shows that the usual constraint qualification conditions ensuring the regularity of the set-valued maps expressing feasibility in optimization problems, are in fact minimal assumptions. These results are then used to derive calculus rules for second-order tangent sets, allowing us in turn to obtain a second-order (Lagrangian) necessary condition for optimality which completes the usual one of positive semidefiniteness on the Hessian of the Lagrangian function.On leave from Universidad de Chile, Casilla 170/3 Correo 3, Santiago, Chile.     

Optimality Conditions for C 1,1 Constrained Multiobjective Problems

The present paper is concerned with the study of the optimality conditions for constrained multiobjective programming problems in which the data have locally Lipschitz Jacobian maps. Second-order necessary and sufficient conditions for efficient solutions are established in terms of second-order subdifferentials of vector functions.     

集值优化Benson真有效元的二阶刻画

引进了一种新的二阶组合切锥, 利用它引进了一种新的二阶组合切导数, 称为二阶组合径向切导数, 并讨论了它的性质及它与二阶组合切导数的关系, 借助二阶径向组合切导数, 分别建立了集值优化取得Benson真有效元的最优性充分和必要条件.