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

Summary: We consider a semi-preinvex programming as follows:

inff(x),subjecttoxKX,g(x)-D,(P)

where K is a semi-connected subset; f:K(Y,C) and g:K(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: γ:[0,1]KX with γ(0)=0 and γ ' (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:
90C48Programming in abstract spaces
49J40Variational methods including variational inequalities
90C25Convex programming
26A51Convexity, generalizations (one real variable)