Virtuous smoothing for global optimization

In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D’Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics, aimed at functions having some limited non-smoothness. Our results pertain to root functions (w^p with 0 < p< 1) and their increasing concave relatives. We provide (i) a sufficient condition (which applies to functions more general than root functions) for our smoothing to be increasing and concave, (ii) a proof that when p= 1 / q for integers q≥ 2 , our smoothing lower bounds the root function, (iii) substantial progress (i.e., a proof for integers 2 ≤ q≤ 10 , 000) on the conjecture that our smoothing is a sharper bound on the root function than the natural and simpler “shifted root function”, and (iv) for all root functions, a quantification of the superiority (in an average sense) of our smoothing versus the shifted root function near 0.