通过幂变换与高斯平滑，加速求解非可微全局优化问题。
Global Optimization with A Power-Transformed Objective and Gaussian Smoothing
- 先对目标函数做指数幂变换，再用高斯平滑处理。
- 在σ∈(0,1)时收敛速度达O(d²σ⁴ε⁻²)，优于标准方法。
- 适合处理非光滑、难优化的复杂目标函数问题。
我们提出一种新方法，分两步求解全局优化问题：(1) 对未必可微的目标函数 $f$ 进行指数幂-$N$ 变换，得到 $f_N$；(2) 使用随机近似优化高斯平滑后的 $f_N$。在 $f$ 满足弱条件下，对任意 $δ>0$，存在足够大的幂 $N_δ$，使该方法收敛至 $f$ 全局最优点的 $δ$-邻域内。收敛率为 $O(d^2σ^4ε^{-2})$，当 $σ$ 预设在 $(0,1)$ 范围时，快于标准及单循环同伦法。多数实验中，本方法所得解优于其他采用平滑技术的算法。
原文摘要 · Abstract (English)
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.
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