向量最优化问题弱解的刻画
Characterizing weak solutions for vector optimization problems
摘要：
本文给出了优化问题的弱解的特征，其中给定的从决策空间$X$到目标空间$Y$的向量函数$F,$，在$x\in C$（其中$C\subset X$是给定的非空约束集）的元素集上“最小化”，满足$G\left( x\right) \leqq_{S}0_{Z},$其中$G$是另一个给定的从$X $到具有正锥$S$的约束空间$Z$的向量函数.这三个空间$X,Y,$和$Z$是局部凸Hausdorff拓扑向量空间，其中$Y$和$Z$分别被两个凸锥$K$和$S,$偏序，并被最大和最小元素扩大。为了得到允许获得根据数据表达的最优性条件的法卡斯引理的合适版本，三元组$\left( F,G,C\right) ,$我们使用$F+I_{A},$的共轭函数的$K-$ epigraph的非渐近表示，其中$I_{A}$表示可行集$A,$的指示函数，即，将$Y$的零向量与$A$的任何元素相关联以及将$Y$的最大元素与$X\diagdown A.$的任何元素相关联的函数
英文摘要：
This paper provides characterizations of the weak solutions of optimization problems where a given vector function $F,$ from a decision space $X$ to an objective space $Y$, is "minimized" on the set of elements $x\in C$ (where $C\subset X$ is a given nonempty constraint set), satisfying $G\left( x\right) \leqq_{S}0_{Z},$ where $G$ is another given vector function from $X $ to a constraint space $Z$ with positive cone $S$. The three spaces $X,Y,$ and $Z$ are locally convex Hausdorff topological vector spaces, with $Y$ and $Z$ partially ordered by two convex cones $K$ and $S,$ respectively, and enlarged with a greatest and a smallest element. In order to get suitable versions of the Farkas lemma allowing to obtain optimality conditions expressed in terms of the data, the triplet $\left( F,G,C\right) ,$ we use non-asymptotic representations of the $K-$epigraph of the conjugate function of $F+I_{A},$ where $I_{A}$ denotes the indicator function of the feasible set $A,$ that is, the function associating the zero vector of $Y$ to any element of $A$ and the greatest element of $Y$ to any element of $X\diagdown A.$
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