Parameter Inversion Using Unconstrained Nonlinear Optimization Methods

Unconstrained nonlinear optimization problems are widely encountered in engineering and scientific computing, with the core objective of finding a set of parameters that extremizes an objective function without considering constraints. Powell's method, as a classical direct search approach, is particularly favored for its gradient-free nature, making it especially suitable for cases where the objective function is non-differentiable or difficult to differentiate. The generalized least squares algorithm is commonly used for parameter estimation and data fitting, optimizing model parameters by minimizing the sum of squared errors.

The core concept of Powell's method involves iteratively constructing conjugate directions to approximate the optimal solution. Its advantage lies in not relying on derivative information of the objective function, instead progressively updating search directions through one-dimensional line searches. Each iteration discards an old direction and adds a new, improved search direction, thereby maintaining conjugacy. While this strategy has slower convergence compared to gradient-based methods, it demonstrates higher stability when handling complex nonlinear functions.

The generalized least squares algorithm addresses model parameter inversion problems by minimizing the weighted residual sum of squares. Unlike ordinary least squares, the generalized approach allows weighting of error terms or incorporation of regularization, thereby enhancing the robustness of parameter estimation. Its implementation typically relies on iteratively reweighted techniques or matrix decomposition methods, such as using QR decomposition or singular value decomposition (SVD) to solve normal equations.

When implementing these algorithms in MATLAB, Powell's method can be achieved by combining loop structures with one-dimensional search functions like `fminbnd`, while generalized least squares can utilize built-in functions like `lsqnonlin` or custom numerical solutions with weighting matrices. Practical applications require attention to initial point selection, iteration termination criteria (such as function value change thresholds or maximum iteration counts), and numerical stability issues (like handling ill-conditioned matrices). Implementation typically involves creating convergence checks and adaptive step size controls within the optimization loops.

The combination of these two methods can be extended to more complex inverse problem solving. For example, in dynamic system parameter identification or inverse modeling, Powell's method can first roughly adjust parameter ranges, followed by refined fitting through generalized least squares. This hierarchical optimization strategy effectively balances global convergence with local accuracy, often implemented through nested optimization loops with switching criteria based on residual thresholds.