Is a convex, nondecreasing function of an invex function invex?

Question

Is a convex, nondecreasing function of an invex function invex?

More broadly, where can I find a list of special properties of functions of invex functions?

Answer 1

Let $h:\mathbb{R} \mapsto \mathbb{R}$ be the convex function, then by sub-gradient inequality and invexity, $$h(f(x)) \geq h(f(u)) + \Delta h(u)(f(x) - f(u)) \geq h(f(u)) + \Delta h(f(u))\langle \Delta f(u), g(x, u) \rangle$$ If $k:\mathbb{R}^n \mapsto \mathbb{R}$ be such that $k(x) = h(f(x))$, then from above, $$k(x) \geq k(u) + \langle \Delta k(u), g(x, u) \rangle$$ So, looks like $k$ will be invex.