Global Optimization with A Power-Transformed Objective and Gaussian Smoothing

TL;DR

This paper introduces a two-step global optimization method using a power transformation and Gaussian smoothing, achieving faster convergence and better solutions compared to existing smoothing-based algorithms.

Contribution

It presents a novel approach combining power transformation with Gaussian smoothing, with theoretical convergence guarantees and improved empirical performance.

Findings

Convergence to near-global solutions under mild conditions.

Faster convergence rate than standard methods when choosing appropriate parameters.

Empirical results show superior solution quality over existing smoothing techniques.

Abstract

We propose a novel method that solves global optimization problems in two steps: (1) perform a (exponential) power-$N$ transformation to the not-necessarily differentiable objective function $f$ and get $f_N$, and (2) optimize the Gaussian-smoothed $f_N$ with stochastic approximations. Under mild conditions on $f$, for any $\delta>0$, we prove that with a sufficiently large power $N_\delta$, this method converges to a solution in the $\delta$-neighborhood of $f$'s global optimum point. The convergence rate is $O(d^2\sigma^4\varepsilon^{-2})$, which is faster than both the standard and single-loop homotopy methods if $\sigma$ is pre-selected to be in $(0,1)$. In most of the experiments performed, our method produces better solutions than other algorithms that also apply smoothing techniques.