More Virtuous Smoothing

In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions $f$ that satisfy $f(0)=0$, $f$ is increasing and concave on $[0,+\mathrm{\infty })$, $f$ is twice differentiable on all of $(0,+\mathrm{\infty })$, but $f^{\mathrm{\prime }}(0)$ is undefined or intolerably large. The canonical examples are root functions $f(w):=w^p$, for $0<p<1$. We consider the earlier approach of defining a smoothing function $g$ that is identical with $f$ on $(\delta ,+\mathrm{\infty })$, for some chosen $\delta >0$, then replacing the part of $f$ on $[0,\delta ]$ with the unique homogeneous cubic, matching $f$, $f^{\mathrm{\prime }}$ and $f^{\mathrm{\prime }\mathrm{\prime }}$ at $\delta$. The parameter $\delta$ is used to control (i.e., upper bound) the derivative at 0 (which controls it on all of $[0,+\mathrm{\infty })$ when $g$ is concave). Our main results: (i) we weaken an earlier sufficient condition to give a necessary and sufficient condition for the piecewise function $g$ to be increasing and concave; (ii) we give a general sufficient condition for $g^{\mathrm{\prime }}(0)$ to be decreasing in the smoothing parameter $\delta$; under the same condition, we demonstrate that the worst-case error of $g$ as an estimate of $f$ is increasing in $\delta$; (iii) we give a general sufficient condition for $g$ to underestimate $f$; (iv) we give a general sufficient condition for $g$ to dominate the simple shift smoothing' $h(w):=f(w+\lambda)-f(\lambda)$ ($\lambda>0$), when the parameters $\delta$ and $\lambda$ are chosen fairly’ — i.e., so that $g^{\mathrm{\prime }}(0)=h^{\mathrm{\prime }}(0)$. In doing so, we solve two natural open problems of Lee and Skipper (2016), concerning (iii) and (iv) for root functions.