More Virtuous Smoothing

Abstract

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 $[0,+\infty)$, $f$ is twice differentiable on all of $(0,+\infty)$, but $f'(0)$ 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 $(\delta,+\infty)$, for some chosen $\delta>0$, then replacing the part of $f$ on $[0,\delta]$ with the unique homogeneous cubic, matching $f$, $f'$ and $f''$ at $\delta$. The parameter $\delta$ is used to control (i.e., upper bound) the derivative at 0 (which controls it on all of $[0,+\infty)$ 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 $g'(0)$ 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 $g'(0)=h'(0)$. In doing so, we solve two natural open problems of Lee and Skipper (2016), concerning (iii) and (iv) for root functions.