MATLAB Implementation of Interior Point Method for Optimization

The interior point method is a numerical algorithm designed for solving linear or nonlinear optimization problems. Its core concept involves iterating within the feasible region to approach the optimal solution, in contrast to the simplex method which moves along the boundary. Implementing interior point methods in MATLAB typically involves these key steps:

Problem formulation: Convert optimization problems into standard form, such as minimizing an objective function f(x) subject to constraints Ax ≤ b. For linear programming problems, both the objective function and constraints must be linear. In MATLAB code, this involves defining objective function handles and constraint matrices.

Barrier function introduction: Interior point methods incorporate barrier functions (like logarithmic barrier functions) to replace inequality constraints, thereby transforming constrained problems into unconstrained or equality-constrained problems. The barrier function ensures iterative points remain inside the feasible region. MATLAB implementation typically uses log-barrier terms like -μ·Σlog(b_i - a_i^T x).

Iterative solution: Use Newton's method or other optimization algorithms to solve equation systems formed by barrier functions and optimality conditions. Each iteration adjusts the barrier parameter μ, gradually tightening the approximation to the feasible region until convergence to the optimal solution. Code implementation involves calculating Newton steps using Hessian matrices and solving linear systems.

Termination conditions: Establish reasonable stopping criteria, such as when the objective function change falls below a threshold or when the barrier parameter becomes sufficiently small, ensuring finite algorithm termination. Common MATLAB implementations check relative function changes and constraint violations.

In MATLAB, you can utilize built-in optimization toolbox functions like `fmincon` with interior-point algorithm options, or manually code the iterative logic. Manual implementation requires handling Hessian matrix computations, line search strategies, and parameter tuning. Key MATLAB functions involved may include `fmincon`, `quadprog`, or custom Newton-step calculations with backslash operators for linear solves.

Interior point methods are particularly suitable for large-scale optimization problems due to their relatively low computational complexity and effective handling of sparse matrix structures. For linear programming, MATLAB's `linprog` function provides interior-point options that can be directly called using algorithm specifications.

For further extension, researchers can investigate dual problem formulations, convergence analysis, or combine interior point methods with other optimization techniques like quasi-Newton methods to improve efficiency. MATLAB's symbolic toolbox can assist in analytical derivative calculations for more complex implementations.