Steepest Descent Method MATLAB Implementation

The steepest descent method is a classical optimization algorithm used to solve unconstrained optimization problems. Its core concept involves iterating along the negative gradient direction of the objective function to gradually approach the minimum point. MATLAB, as a crucial tool in scientific computing, is particularly well-suited for implementing such numerical optimization algorithms.

In MATLAB implementation, the steepest descent method typically involves these key steps: First, compute the gradient at the current point using MATLAB's gradient functions or symbolic differentiation. Then determine the step size through strategies like exact line search or Armijo condition, often implemented using while-loops for iterative refinement. Finally, update the iteration point using vector operations. This iterative process continues until meeting preset convergence criteria, such as when the gradient norm falls below a specified threshold, which can be monitored using norm() functions.

The conjugate gradient method represents a more efficient optimization algorithm that accelerates convergence by constructing conjugate directions. Compared to the steepest descent method, conjugate gradient utilizes historical information during iteration to avoid "zigzagging phenomena," significantly improving convergence speed. A typical MATLAB implementation includes three core modules: direction updating using mathematical recurrence relations, step size calculation with dot product operations, and convergence assessment with conditional statements.

International textbook implementations often emphasize algorithm robustness and generality, typically incorporating adaptive step size strategies and stricter convergence criteria. These code examples provide valuable references for understanding engineering implementations of optimization algorithms. Mastering MATLAB implementations of these classical optimization algorithms represents valuable skills for both academic research and engineering applications, particularly for developing custom optimization solvers and performance benchmarking.