Optimality conditions for semi-preinvex programming.

*(English)*Zbl 0894.90164Summary: We consider a semi-preinvex programming as follows:

$$inff\left(x\right),\phantom{\rule{1.em}{0ex}}\text{subject}\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{4.pt}{0ex}}x\in K\subseteq X,\phantom{\rule{4pt}{0ex}}g\left(x\right)\in -D,\phantom{\rule{2.em}{0ex}}\left(\mathrm{P}\right)$$

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 $\gamma \left(0\right)=0$ and ${\gamma}^{\text{'}}\left({0}^{+}\right)=u$, 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.