Global Optimization with A Power-Transformed Objective and Gaussian Smoothing
This addresses global optimization problems for researchers and practitioners, offering a novel method with proven convergence, though it appears incremental as it builds on smoothing techniques.
The paper tackles global optimization of non-differentiable functions by applying a power transformation and Gaussian smoothing, proving convergence to a δ-neighborhood of the global optimum with a rate of O(d²σ⁴ε⁻²) and showing better experimental results than other smoothing-based algorithms.
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 $δ>0$, we prove that with a sufficiently large power $N_δ$, this method converges to a solution in the $δ$-neighborhood of $f$'s global optimum point. The convergence rate is $O(d^2σ^4\varepsilon^{-2})$, which is faster than both the standard and single-loop homotopy methods if $σ$ 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.