Mathematical programming with a class of non-smooth functions (Q1607572)

The authors use the following definition: The function \(f: \mathbb{R}^n \to\mathbb{R}\) is said to be sub-invex at \(x\in \mathbb{R}^n\) with respect to vector function \(\eta:\mathbb{R}^n \times\mathbb{R}^n \to\mathbb{R}^n\) if there exists an element \(\xi\in \mathbb{R}^n\) such that \(f(y)- f(x)\geq \langle\xi,\eta (y,x)\rangle\), \(\forall y\in \mathbb{R}^n\). All nonsmooth invex function with respect to \(\eta\) is a sub-invex function in respect to \(\eta\). The authors consider a nonsmooth scalar program, generated by sub-invex functions with respect to the same \(\eta\) and they establish necessary optimality conditions of Karush-Kuhn-Tucker type for this program. These conditions are used to establish weak and strong duality theorems in Wolfe's sense.