ABSTRACT

In this paper, we investigate some optimality conditions of higher-order for nonsmooth nonconvex vector equilibrium problems with constraints (or without constraints) in terms of the higher-order upper and lower Studniarski derivatives in Banach spaces. The calculus rule of all the data involved in the problem is taken into account. Using the notion of higher-order upper Studniarski derivative with the class of m-stable/m-steady functions, we first provide the higher-order necessary optimality conditions for the local weak efficient solution of vector equilibrium problem without constraints and then we present the higher-order necessary and sufficient optimality conditions for the local strict minimum of order m to such problem. We second obtain the necessary and sufficient optimality conditions for those efficient solutions of vector equilibrium problem with cone and equality constraints through Lagrange multiplier rules in finite-dimensional spaces. Final, the higher-order necessary and sufficient optimality conditions in terms of the higher-order upper and lower Studniarski derivatives for the local weak efficient solution of vector equilibrium problem with set, cone and equality constraints in Banach spaces are also established. Some examples are proposed to demonstrate our findings.

摘要

在本文中，我们根据 Banach 空间中的高阶和低阶 Studniarski 导数研究了有约束（或无约束）的非光滑非凸向量平衡问题的一些高阶最优性条件。考虑到问题中涉及的所有数据的微积分规则。利用具有m -stable / m -stable函数类的高阶上Studniarski导数的概念，我们首先为无约束向量平衡问题的局部弱有效解提供了高阶必要最优性条件，然后我们给出了m阶局部严格最小值的高阶充要最优性条件对这样的问题。其次，我们通过有限维空间中的拉格朗日乘子规则，获得了具有锥约束和等式约束的向量平衡问题的有效解的充要最优性条件。最后，还建立了Banach空间中具有集合、锥和等式约束的向量平衡问题的局部弱有效解的高阶上Studniarski导数和下Studniarski导数的高阶充要最优性条件。提出了一些例子来证明我们的发现。