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首页> 外文期刊>Rendiconti del Circolo Matematico di Palermo >Stability of properly efficient points and isolated minimizers of constrained vector optimization problems
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Stability of properly efficient points and isolated minimizers of constrained vector optimization problems

机译：约束向量优化问题的适当有效点和孤立极小值的稳定性

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In this paper the constrained vector optimization problem mic C f(x), g(x) ? ? K, is considered, where $f:mathbb{R}^n to mathbb{R}^m $ and $g:mathbb{R}^n to mathbb{R}^p $ are locally Lipschitz functions and $C subset mathbb{R}^m $ and $K subset mathbb{R}^p $ are closed convex cones. Several solution concepts are recalled, among them the concept of a properly efficient point (p-minimizer) and an isolated minimizer (i-minimizer). On the base of certain first-order optimalitty conditions it is shown that there is a close relation between the solutions of the constrained problem and some unconstrained problem. This consideration allows to “double” the solution concepts of the given constrained problem, calling sense II optimality concepts for the constrained problem the respective solutions of the related unconstrained problem, retaining the name of sense I concepts for the originally defined optimality solutions. The paper investigates the stability properties of thep-minimizers andi-minimizers. It is shown, that thep-minimizers are stable under perturbations of the cones, while thei-minimizers are stable under perturbations both of the cones and the functions in the data set. Further, it is shown, that sense I concepts are stable under perturbations of the objective data, while sense II concepts are stable under perturbations both of the objective and the constraints. Finally, the so called structural stability is discused.
机译：本文将约束向量优化问题mic C f（x），g（x）？ ？考虑了K，其中$ f：mathbb {R} ^ n到mathbb {R} ^ m $和$ g：mathbb {R} ^ n到mathbb {R} ^ p $是本地Lipschitz函数和$ C子集mathbb {R} ^ m $和$ K子集mathbb {R} ^ p $是封闭的凸锥。召回了几种解决方案概念，其中包括有效点（p-minimizer）和隔离的最小化器（i-minimizer）的概念。在某些一阶最优性条件的基础上，证明了约束问题的解与一些非约束问题的解之间存在密切的关系。这种考虑允许将给定约束问题的解决方案概念“加倍”，将受约束问题的感官II最优性概念称为相关无约束问题的相应解决方案，为最初定义的最优性解决方案保留感官I概念的名称。本文研究了p最小化器和i最小化器的稳定性。结果表明，p最小化子在视锥的扰动下是稳定的，而i最小化子在视锥和数据集的函数的扰动下是稳定的。此外，示出了感觉I概念在目标数据的扰动下是稳定的，而感觉II概念在目标和约束的扰动下是稳定的。最后，讨论了所谓的结构稳定性。

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