ε-真有效解的非线性标量化

Nonlinear Scalarization ofε-Properly Efficient Solutions

本文利用基于点闭凸锥的经典非线性标量化函数Δ-K对向量优化问题ε-真有效解的非线性标量化性质进行了研究。首先证明了向量优化问题(VP)的ε-真有效解蕴含标量化问题(Py)的dε+K(0)-近似解,并通过例子说明了这一结论的逆不一定成立。进一步,证明了标量化问题(Py)的严格β-近似解蕴含向量优化问题(VP)的ε-真有效解,并举例说明了如果集合f(S)+ε+K-f(x)的锥包不是闭集,这一结论不一定成立以及标量化问题(Py)的β-近似解不一定蕴含向量优化问题(VP)的ε-真有效解。

In this paper, we study some nonlinear scalarization characterizations of e-properly efficient solutions for vector optimization problems via the classical nonlinear scalarization function Δ-κ obtained by using a pointed closed convex cone. We first prove that s-properly efficient solutions of the vector optimization problem （VP） implies dε＋n （0）-approximate solutions of the scalarization problem （Py） and give an example to illustrate the fact that the converse of this conclusion may not be valid. Furthermore, we also prove that strictly/9-approximate solutions of the scalarization problem （Py） implies e-properly efficient solutions of the vector optimization problem （VP）, and propose some examples to illustrate the facts that this conclusion may not be true if the cone hull of the set f（S）＋ε＋K-f（^-x） is not closed, and β-approximate solutions of the scalarization problem （Py） does not imply e-properly efficient solutions of the vector optimization problem （VP）.