A Unified Separation Theorem for Closed Sets in a Banach Space and Optimality Conditions for Vector Optimization
A Unified Separation Theorem for Closed Sets in a Banach Space and Optimality Conditions for Vector Optimization
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DOI：
10.1137/100811155
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发表时间：
2011-09
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影响因子：
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通讯作者：
Xi Yin Zheng;K. Ng
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文献类型：
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作者：
Xi Yin Zheng;K. Ng
利用变分分析技术，根据法锥，我们对Banach空间中的有限多个闭（不一定是凸）集建立了统一的分离结果，不仅覆盖了现有的非凸分离结果和经典的凸分离定理，而且重新捕获了近似投影定理。借助闭集的分离结果，我们为约束向量优化问题的近似Pareto解提供了充分必要条件。特别是，我们将数值优化问题近似解的一些基本最优性结果扩展到向量优化设置。
Using the technique of variational analysis and in terms of normal cones, we establish unified separation results for finitely many closed (not necessarily convex) sets in Banach spaces, which not only cover the existing nonconvex separation results and a classical convex separation theorem, but also recapture the approximate projection theorem. With help of the separation result for closed sets, we provide necessary and sufficient conditions for approximate Pareto solutions of constrained vector optimization problems. In particular, we extend some basic optimality results for approximate solutions of numerical optimization problems to the vector optimization setting.