Global Optimization with a Power-Transformed Objective and Gaussian Smoothing

Abstract

We propose a novel method, namely Gaussian Smoothing with a Power-Transformed Objective (GS-PowerOpt), that solves global optimization problems in two steps: (1) perform a (exponential) power-$N$ transformation to the not necessarily differentiable objective $f:\mathbb{R}^d\rightarrow \mathbb{R}$ 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, at the iteration complexity of $O(d^4\varepsilon^{-2})$. If we require that $f$ is differentiable and further assume the Lipschitz condition on $f$ and its gradient, the iteration complexity reduces to $O(d^2\varepsilon^{-2})$, which is significantly faster than the standard homotopy method. In most of the experiments performed, our method produces better solutions than other algorithms that also apply the smoothing technique.