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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
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