More Virtuous Smoothing

TL;DR

This paper develops a refined smoothing technique for univariate functions used in mixed-integer nonlinear optimization, providing necessary and sufficient conditions for properties like monotonicity, concavity, and underestimation, especially for root functions.

Contribution

It introduces a weakened, necessary and sufficient condition for the smoothing function to be increasing and concave, and addresses open problems related to root functions from prior research.

Findings

Provides a necessary and sufficient condition for the smoothing function to be increasing and concave.

Shows that the derivative of the smoothing function at zero decreases with the smoothing parameter under certain conditions.

Establishes conditions under which the smoothing function underestimates the original function.

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.