In the field of global optimization, many existing algorithms face challenges posed by non-convex target functions and high computational complexity or unavailability of gradient information. These limitations, exacerbated by sensitivity to initial conditions, often lead to suboptimal solutions or failed convergence. This is true even for Metaheuristic algorithms designed to amalgamate different optimization techniques to improve their efficiency and robustness. To address these challenges, we develop a sequence of multidimensional integration-based methods that we show to converge to the global optima under some mild regularity conditions. Our probabilistic approach does not require the use of gradients and is underpinned by a mathematically rigorous convergence framework anchored in the nuanced properties of nascent optima distribution. In order to alleviate the problem of multidimensional integration, we develop a latent slice sampler that enjoys a geometric rate of convergence in generating samples from the nascent optima distribution, which is used to approximate the global optima. The proposed Probabilistic Global Optimizer (ProGO) provides a scalable unified framework to approximate the global optima of any continuous function defined on a domain of arbitrary dimension. Empirical illustrations of ProGO across a variety of popular non-convex test functions (having finite global optima) reveal that the proposed algorithm outperforms, by order of magnitude, many existing state-of-the-art methods, including gradient-based, zeroth-order gradient-free, and some Bayesian Optimization methods, in term regret value and speed of convergence. It is, however, to be noted that our approach may not be suitable for functions that are expensive to compute.

翻译：在全球优化领域，许多现有算法面临非凸目标函数带来的挑战，以及高计算复杂度或梯度信息不可用的问题。这些局限性，加之对初始条件的敏感性，往往导致次优解或收敛失败。即使是为融合不同优化技术以提高效率和鲁棒性而设计的元启发式算法，也存在这一情况。为应对这些挑战，我们开发了一系列基于多维积分的方法，并证明其在温和正则条件下收敛到全局最优解。我们的概率方法无需使用梯度，并建立在严格数学收敛框架之上，该框架依赖于新生最优分布的细微性质。为缓解多维积分问题，我们提出了一种隐式切片采样器，该采样器在从新生最优分布生成样本时具有几何收敛速度，用于逼近全局最优解。所提出的概率全局优化器（ProGO）提供了一个可扩展的统一框架，用于逼近定义在任意维度域上的任何连续函数的全局最优解。ProGO在多种具有有限全局最优解的流行非凸测试函数上的实证结果表明，与许多现有最先进方法（包括基于梯度的方法、零阶无梯度方法以及部分贝叶斯优化方法）相比，所提算法在遗憾值和收敛速度方面提升了一个数量级。但需注意，我们的方法可能不适用于计算代价高昂的函数。