Interior Point Method Implementation Using MATLAB Code

The interior point method is a numerical algorithm used to solve linear or nonlinear optimization problems, characterized by iterating within the feasible region to approach the optimal solution. MATLAB, as a commonly used tool in numerical computing, is particularly well-suited for implementing this type of algorithm.

The core algorithmic approach can be divided into three key steps:

Initial Point Selection: The interior point method requires starting iterations from a strictly feasible interior point. This initial point must satisfy all constraints and must not lie on the boundary. In MATLAB implementation, this typically involves using optimization functions like fmincon to find a feasible starting point or employing specialized initialization routines.

Barrier Function Construction: By introducing logarithmic barrier functions, constraint conditions are integrated into the objective function, transforming the original constrained problem into a series of unconstrained subproblems. MATLAB implementation uses functions like log() to create barrier terms and combines them with the original objective function using element-wise operations.

Central Path Following: Newton's method is used to iteratively solve these subproblems, gradually approaching the optimal solution of the original problem by adjusting barrier parameters. In MATLAB, this involves calculating Hessian matrices and gradients using symbolic computation or numerical differentiation, then solving linear systems with backslash operator or linsolve function.

Several technical details require attention during MATLAB implementation:

Using sparse matrix storage can significantly improve computational efficiency for large-scale problems through functions like sparse() and speye() Appropriate setting of barrier parameter reduction coefficients crucially affects convergence speed, typically implemented using geometric progression with carefully chosen ratios Effective step size selection strategies must be designed to ensure iterative points remain within the feasible domain, often using line search algorithms with feasibility checks

The main advantage of this algorithm compared to the simplex method lies in its handling capability for large-scale problems. Particularly when problem dimensions are high, the interior point method generally demonstrates better computational efficiency. In practical applications, performance comparison and validation can be conducted using MATLAB's Optimization Toolbox functions such as interior-point algorithm options in fmincon.