New Optimality Conditions for a Nondifferentiable Fractional Semipreinvex Programming Problem

Abstract

We study a nondifferentiable fractional programming problem as follows: $(P){\min }_{x\in K}f(x)/g(x)$ subject to $x\in K\subseteq X,h_i(x)\le 0,i=\mathrm{1,2},\dots ,m$, where $K$ is a semiconnected subset in a locally convex topological vector space $X$, $f:K\to ℝ$, $g:K\to {ℝ}_{+}$ and $h_i:K\to ℝ$, $i=\mathrm{1,2},\dots ,m$. If $f$, $-g$, and $h_i$, $i=\mathrm{1,2},\dots ,m$, are arc-directionally differentiable, semipreinvex maps with respect to a continuous map $\gamma :[\mathrm{0,1}]\to K\subseteq X$ satisfying $\gamma (0)=0$ and ${\gamma }^{\prime }(0^{+})\in K$, then the necessary and sufficient conditions for optimality of $(P)$ are established.