Virtuous smoothing for global optimization

TL;DR

This paper introduces a new cubic smoothing technique for root functions in global optimization, providing theoretical guarantees and demonstrating its superiority over existing methods near zero.

Contribution

Develops a virtuous cubic smoothing method for root functions, with theoretical conditions and proofs of its bounds and superiority over shifted root functions.

Findings

Smoothing is increasing and concave under certain conditions.

When p=1/q, smoothing lower bounds the root function.

Proves smoothing is sharper than shifted root function for q up to 10,000.

Abstract

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 $q\geq 2$, our smoothing lower bounds the root function, (iii) substantial progress (i.e., a proof for integers $2\leq q\leq 10,000$) 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.