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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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