Second-order contingent derivatives of set-valued mappings with application to set-valued optimization

In this paper, a family of parameterized set-valued optimization problems, whose constraint set depends on a parameter, are considered. Some calculus rules are obtained for calculating the second-order contingent derivatives of the composition and sum of two set-valued mappings. Then, by using these calculus rules, some results concerning second-order sensitivity analysis are established, and an explicit expression for the second-order contingent derivative of the (weak) perturbation mapping in the set-valued optimization problems is obtained.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

Optimality Conditions for Vector Equilibrium Problems in Terms of Contingent Epiderivatives

In this article, we study some important properties of contingent epiderivatives concerning steady functions and a cone with a compact base along with its applications to establish necessary and sufficient optimality conditions for weakly efficient, Henig efficient, globally efficient and superefficient solutions for no constraints and constraints (it concludes cone constraint, equality constraint and a constraint set) vector equilibrium problems in terms of contingent epiderivatives. We also give some examples to illustrate obtained results.     

Higher-order optimality conditions for strict local minima

In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint. We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two, using now a special second order derivative which leads to the Fréchet derivative in the differentiable case.     

On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and C ^1,1 functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.     

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.     

Nonsmooth semi-infinite programming problems with mixed constraints

We consider a nonsmooth semi-infinite programming problem with a feasible set defined by inequality and equality constraints and a set constraint. First, we study some alternative theorems which involve linear and sublinear functions and a convex set and we propose several generalizations of them. Then, alternative theorems are applied to obtain, under different constraint qualifications, several necessary optimality conditions in the type of Fritz-John and Karush-Kuhn-Tucker.     

Contingent derivatives of set-valued maps and applications to vector optimization

In this paper we investigate contingent derivatives of set-valued maps and their lower and upper semidifferentiability properties. We provide also some calculus rules for these derivatives in infinite dimensional spaces. The concept of contingent derivatives is then applied to produce several necessary and sufficient conditions for vector optimization problems with set-valued objectives.This paper was written when the author was at the University of Erlangen-Nurnberg under a grant of the Alexander von Humboldt Foundation.On leave from the Institute of Mathematics, Hanoi, Vietnam.     

集值映射的次微分的性质及其应用(英文)

本文讨论了文章"Subgradient of S-convex set-valued mappings and weak efficientsolutions"(Appl.Math.J.Chinese Univ.1998,13(4):463-472)中引入的集值映射的次微分的性质及应用.利用相依导数的性质,讨论次微分的性质,并得到了两个集值映射的和、复合以及交的次微分的运算法则.最后,通过这种次微分得到了集值优化问题最优性条件的充要条件,同时推广了此文中的定理7.     

Necessary Optimality Conditions in Terms of Convexificators in Lipschitz Optimization

This study is devoted to constraint qualifications and Kuhn-Tucker type necessary optimality conditions for nonsmooth optimization problems involving locally Lipschitz functions. The main tool of the study is the concept of convexificators. First, the case of a minimization problem in the presence of an arbitrary set constraint is considered by using the contingent cone and the adjacent cone to the constraint set. Then, in the case of a minimization problem with inequality constraints, Abadie type constraint qualifications and several other qualifications are proposed; Kuhn-Tucker type necessary optimality conditions are derived under the qualifications.Communicated by S. SchaibleThe authors thank the referees for bringing to their attention some papers closely related to this study and for helpful comments and constructive suggestions that have greatly improved the original version of the paper. Further, they are indebted to Professors H. W. Sun and F. Y. Lu, who suggested an example for this paper. The first author thanks S. Schaible for encouragement during this research.     

Codifferential Calculus

In this paper, some exact calculus rules are obtained for calculating the coderivatives of the composition of two multivalued maps. Similar rules are displayed for sums. A crucial role is played by an intermediate set-valued map called the resolvent. We first establish inclusions for contingent, Fréchet and limiting coderivatives. Combining them, we get equality rules. The qualification conditions we present are natural and less exacting than classical conditions.     

集值映射的次微分的性质及其应用(英)

本文讨论了文章``Subgradient of S-convex set-valued mappings and weak efficient solutions"(Appl.Math. J. Chinese Univ. 1998, 13(4): 463-472) 中引入的集值映射的次微分的性质及应用.利用相依导数的性质,讨论次微分的性质,并得到了两个集值映射的和、复合以及交的次微分的运算法则.最后,通过这种次微分得到了集值优化问题最优性条件的充要条件,同时推广了此文中的定理7.     

Some Calculus Rules for Contingent Epiderivatives

This article presents some calculus rules for contingent epiderivatives of set-valued maps. Among other results the main emphasis is focused on a formula for scalar multiplication, sum formulae and chain rules. The calculus of contingent cones and some inversion theorems are used as a tool. Some applications are also given.     

Optimality conditions in optimization problems with convex feasible set using convexificators

In this paper, we consider a nonsmooth optimization problem with a convex feasible set described by constraint functions which are neither convex nor differentiable nor locally Lipschitz necessarily. Utilizing upper regular convexificators, we characterize the normal cone of the feasible set and derive KKT type necessary and sufficient optimality conditions. Under some assumptions, we show that the set of KKT multipliers is bounded. We also characterize the set of optimal solutions and introduce a linear approximation corresponding to the original problem which is useful in checking optimality. The obtained outcomes extend various results existing in the literature to a more general setting.     

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.     

Generalized second-order directional derivatives and optimization with C1,1 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C¹,¹ functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C^1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.     

Necessary and Sufficient Conditions for Weak Quasi Efficiency in Nonsmooth Multiobjective Problems

In this article, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given and the relations between them are analyzed. We establish Kuhn-Tucker and strong Kuhn-Tucker necessary optimality conditions for (weak) quasi e?ciency in terms of the Clarke subdifferential. By using two new classes of generalized convex functions, su?cient conditions for local (weak) quasi e?cient are also provided. Furthermore, we study the Mond-Weir type dual problem and establish weak, strong and converse duality results.     

A characterization of paratingent cone andP-subderivative with applications in nonsmooth analysis

We give a new characterization of the paratingent cone in terms of contingent cones, i.e., the paratingent cone to any open set at a boundary point is the upper limit of the contingent cones at the neighboring points. We use this result to characterize the strict differentiability in terms of the contingent directional derivatives. We also define aP-subderivative for continuous functions and develop a subdifferential calculus with applications to optimality conditions in mathematical programming.     

集值映射多目标规划的K-T最优性条件

讨论集值映射多目标规划(VP)的最优性条件问题.首先,在没有锥凹的假设下,利用集值映射的相依导数,得到了(VP)的锥--超有效解要满足的必要条件和充分条件.其次,在锥凹假设和比推广了的Slater规格更弱的条件下,给出了(VP)关于锥--超有效解的K--T型最优性必要条件和充分条件.     

Differential properties and optimality conditions for generalized weak vector variational inequalities

In this paper, we study a generalized weak vector variational inequality, which is a generalization of a weak vector variational inequality and a Minty weak vector variational inequality. By virtue of a contingent derivative and a Φ-contingent cone, we investigate differential properties of a class of set-valued maps and obtain an explicit expression of its contingent derivative. We also establish some necessary optimality conditions for solutions of the generalized weak vector variational inequality, which generalize the corresponding results in the literature. Furthermore, we establish some unified necessary and sufficient optimality conditions for local optimal solutions of the generalized weak vector variational inequality. Simultaneously, we also show that there is no gap between the necessary and sufficient conditions under an appropriate condition.