## New optimality conditions for a nondifferentiable fractional semipreinvex programming problem.(English)Zbl 1266.90146

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

### MSC:

 90C26 Nonconvex programming, global optimization
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### References:

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