An Efficient Dai-Yuan CG Method Based on Structured Secant Conditions for NLS Problems and Its Application

Nonlinear least squares (NLS) problems are commonly encountered in a wide range of scientific and engineering fields, often demanding robust and efficient optimization methods for their solution. Traditional approaches for solving NLS problems frequently encounter difficulties related to high computational costs and memory usage, particularly in large-scale scenarios. This study introduces a novel structured Dai-Yuan two-term conjugate gradient (CG) method to address the minimization of NLS problems. By leveraging a Taylor series expansion of the objective function's Hessian, a structured vector approximation, representing a matrix-vector interaction, is constructed. This formulation adheres to a quasi-Newton condition. The method utilizes this structured approximation to enrich the conventional search direction with additional curvature information from the Hessian. The resulting search direction satisfies the required descent condition. Furthermore, numerical experiments on a range of benchmark problems demonstrate that the proposed method offers superior performance, outperforming several existing techniques.