The author unifies various definitions of non-smooth invex functions by introducing the \(K\)-directional derivative of a function in the following way. Let \(X\subset\mathbb{R}^n\) be an open set, \(f:X\to\mathbb{R}\), \(x\in X\) and \(K\) be a local cone approximation; the positive homogeneous function \(f^K(x,.): \mathbb{R}^n\to [-\infty,\infty]\) defined by \(f^K(x,y)= \inf\{\beta \in\mathbb{R}: (y, \beta)\in K(\text{epi} (x,f(x))\}\) is called the \(K\)-directional derivative of \(f\) at \(x\). Characterization of a \(K\)-inf-stationary point for \(K\)-quasidifferentiable functions and \(K\)-invexity for \(K\)-subdifferentiable functions are given. Sufficient optimality conditions are obtained for a nonlinear programming problem in which the objective function \(f_0\) is \(K_0\)-pseudoinvex and constraint functions \(f_i\) are \(K_i\)-quasiinvex. The well known weak and strong duality theorems are also established.