Powell Dogleg Method for Solving Trust-Region Subproblems

In computational mathematics, the Powell dogleg method is an optimization algorithm designed for solving trust-region subproblems. Primarily applied to nonlinear least-squares problems, this method is widely used in numerical optimization and nonlinear optimization scenarios within computer graphics. The algorithm effectively combines the advantages of the steepest descent method and Newton's method, ensuring convergence and global optimal solutions during computation. Consequently, it has gained extensive practical applications in optimization problems across engineering design and scientific research domains. The algorithm implementation typically involves three key computational steps: first calculating the Cauchy point (steepest descent direction), then computing the Newton point, and finally determining the optimal solution along the dogleg path constrained by the trust-region radius. Code implementation often includes functions for Hessian matrix computation, trust-region radius adjustment, and dogleg path interpolation. In trust-region subproblems, the algorithm enhances computational efficiency and precision by utilizing the dogleg path to locate optimal solutions, with common termination criteria including gradient norms and trust-region boundary checks.