Scalable subspace methods for derivative-free nonlinear least-squares optimization
提出一种基于随机子空间迭代的模型构建框架，用于大规模无导数优化，并专门针对非线性最小二乘问题设计了DFBGN算法，通过局部线性插值近似雅可比矩阵，在用户指定维度的子空间中计算新步长，实现了低线性代数成本和高可扩展性。
Abstract We introduce a general framework for large-scale model-based derivative-free optimization based on iterative minimization within random subspaces. We present a probabilistic worst-case complexity analysis for our method, where in particular we prove high-probability bounds on the number of iterations before a given optimality is achieved. This framework is specialized to nonlinear least-squares problems, with a model-based framework based on the Gauss–Newton method. This method achieves scalability by constructing local linear interpolation models to approximate the Jacobian, and computes new steps at each iteration in a subspace with user-determined dimension. We then describe a practical implementation of this framework, which we call DFBGN. We outline efficient techniques for selecting the interpolation points and search subspace, yielding an implementation that has a low per-iteration linear algebra cost (linear in the problem dimension) while also achieving fast objective decrease as measured by evaluations. Extensive numerical results demonstrate that DFBGN has improved scalability, yielding strong performance on large-scale nonlinear least-squares problems.