## Optimality conditions for semi-preinvex programming.(English)Zbl 0894.90164

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:

 90C48 Programming in abstract spaces 49J40 Variational inequalities 90C25 Convex programming 26A51 Convexity of real functions in one variable, generalizations
Full Text: