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Zbl 0894.90164
Lai, Hang-Chin
Optimality conditions for semi-preinvex programming.
(English)
[J] Taiwanese J. Math. 1, No.4, 389-404 (1997). ISSN 1027-5487

Summary: We consider a semi-preinvex programming as follows: $$\inf f(x),\quad\text{subject to }x\in K\subseteq X,\ g(x)\in -D,\tag P$$ where $K$ is a semi-connected subset; $f:K\to (Y,C)$ and $g: K\to(Z,D)$ are semi-preinvex maps; while $(Y,C)$ and $(Z,D)$ are ordered vector spaces with order cones $C$ and $D$, respectively. If $f$ and $g$ are arc-directionally differentiable semi-preinvex maps with respect to a continuous map: $\gamma:[0, 1]\to K\subseteq X$ with $\gamma(0)= 0$ and $\gamma'(0^+)= u$, then the necessary and sufficient conditions for optimality of (P) is established. It is also established that a solution of an unconstrained semi-preinvex optimization problem is related to a solution of a semi-prevariational inequality.
MSC 2000:
*90C48 Programming in abstract spaces
49J40 Variational methods including variational inequalities
90C25 Convex programming
26A51 Convexity, generalizations (one real variable)

Keywords: semi-preinvex programming; ordered vector spaces; semi-prevariational inequality

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