Bisection Method in Optimization Algorithms

In numerical computation and optimization problems, the bisection method is a simple yet efficient algorithm used to locate roots or extremum points of functions within continuous intervals. This method employs continuous interval halving to progressively approach the target solution, characterized by stable convergence and straightforward implementation.

The core principle of the bisection method involves: First, defining an initial interval where the function exhibits opposite signs at the endpoints (satisfying the Intermediate Value Theorem conditions). Calculating the midpoint of the interval and evaluating the function value at this point. Adjusting the interval boundaries based on the midpoint's function value by retaining the half-interval containing the root and discarding the other half. Iterating these steps until the interval length falls below a preset tolerance or maximum iteration count is reached, with the final midpoint serving as the approximate solution.

In MATLAB implementation, the algorithm typically integrates loop structures and conditional judgments to ensure efficiency and correctness. Leveraging MATLAB's robust numerical computation capabilities, the bisection method can rapidly handle optimization problems involving complex functions, particularly suited for single-variable continuous function scenarios. Key implementation steps include: using a while-loop for iterative refinement, employing sign comparison functions (e.g., sign()) for root localization, and incorporating tolerance checks for termination conditions.

As a fundamental optimization algorithm, while the bisection method's convergence rate is slower compared to higher-order methods like Newton's approach, its stability ensures broad applicability in practical engineering problems where derivative information is unavailable or function behavior is highly nonlinear.