Characterizations of Local and Global Error Bounds for Convex Inequalities in Banach Spaces
Characterizations of Local and Global Error Bounds for Convex Inequalities in Banach Spaces
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DOI：
10.1137/050644872
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发表时间：
2007-02
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影响因子：
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通讯作者：
Hui Hu
中科院分区：
文献类型：
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作者：
Hui Hu
研究了Banach空间中由真凸函数定义的凸不等式的局部和整体误差界。引入弱基本约束条件（弱BCQ）的概念来控制解集边界点的法线方向。局部和全局误差界的特点是由方向长度分解条件，这提供了一种方法来独立验证弱BCQ和次微分的长度控制。为了进一步刻画全局误差界，提出并研究了分段扩张性质。它示出的方向长度条件的验证全局误差界可以充分进行任何子集具有段扩展属性，而不是整个边界。这导致了一个简单的公式，最小的全局误差界。在欧氏空间中，该条件的验证和最小全局误差界的计算可以在极值点集上进行。
This paper studies local and global error bounds for a convex inequality defined by a proper convex function in a Banach space. The concept of weak basic constraint qualification (weak BCQ) is introduced to control the normal directions at a boundary point of the solution set. Local and global error bounds are characterized by a direction-length decomposition condition, which provides a way to independently verify the weak BCQ and the length control of the subdifferential. To further characterize global error bounds, the segment extension property is proposed and studied. It is shown that the verification of the direction-length condition for global error bounds can be sufficiently carried out on any subset having the segment extension property instead of the entire boundary. This leads to a simple formula for the smallest global error bound. In the Euclidean space, the verification of the condition and the computation of the smallest global error bound can be carried out on the set of extreme points.