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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?

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Please try and make the question's body self-contained. – Asaf Karagila Feb 6 '13 at 2:40
What is an invex function? – copper.hat Feb 6 '13 at 3:45
up vote 0 down vote accepted

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.

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