Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems
Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems
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DOI：
10.1007/s10957-006-9108-6
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发表时间：
2006-12
影响因子：
1.9
通讯作者：
Zili Wu;Soon-Yi Wu
中科院分区：
文献类型：
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作者：
Zili Wu;Soon-Yi Wu
对于赋范向量空间中的凸规划，目标函数在最优解处允许Gâteaux导数，我们证明了解集由位于其法向量等于Gâteaux导数的超平面中的可行点组成.对于一般的连续凸规划，可行点是最优解当且仅当它位于一个超平面内，且法向量属于目标函数在该点的次微分。在几种情况下，证明了变分不等式问题的解集与以对偶间隙函数为目标函数的凸规划的解集是一致的，而所涉及的映射可以用来表示上述法向量.
For a convex program in a normed vector space with the objective function admitting the Gâteaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gâteaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors.