We propose a novel evolutionary algorithm for optimizing real-valued objective functions defined on the Grassmann manifold Gr}(k,n), the space of all k-dimensional linear subspaces of R^n. While existing optimization techniques on Gr}(k,n) predominantly rely on first- or second-order Riemannian methods, these inherently local methods often struggle with nonconvex or multimodal landscapes. To address this limitation, we adapt the Differential Evolution algorithm - a global, population based optimization method - to operate effectively on the Grassmannian. Our approach incorporates adaptive control parameter schemes, and introduces a projection mechanism that maps trial vectors onto the manifold via QR decomposition. The resulting algorithm maintains feasibility with respect to the manifold structure while enabling exploration beyond local neighborhoods. This framework provides a flexible and geometry-aware alternative to classical Riemannian optimization methods and is well-suited to applications in machine learning, signal processing, and low-rank matrix recovery where subspace representations play a central role. We test the methodology on a number of examples of optimization problems on Grassmann manifolds.

翻译：本文提出了一种新颖的进化算法，用于优化定义在格拉斯曼流形 Gr(k,n)（即 R^n 中所有 k 维线性子空间构成的流形）上的实值目标函数。尽管现有的 Gr(k,n) 优化技术主要依赖于一阶或二阶黎曼方法，但这些本质上是局部的方法在处理非凸或多峰优化地形时常常面临困难。为了克服这一局限性，我们将差分进化算法——一种基于种群的全局优化方法——进行改造，使其能在格拉斯曼流形上有效运行。我们的方法融合了自适应控制参数策略，并引入了一种通过 QR 分解将试验向量投影到流形上的机制。所得算法在保持与流形结构相容的可行性的同时，能够探索局部邻域之外的空间。该框架为经典的黎曼优化方法提供了一种灵活且几何感知的替代方案，非常适用于机器学习、信号处理和低秩矩阵恢复等子空间表示起核心作用的应用领域。我们在多个格拉斯曼流形优化问题的实例上测试了该方法。