Solving Inequality-Constrained Convex Optimization Problems Using Barrier Methods and Primal-Dual Interior Point Methods

In this article, we explore how MATLAB employs barrier methods and primal-dual interior point methods to solve inequality-constrained convex optimization problems. The barrier method represents a fundamental optimization technique that transforms constraints into barrier functions incorporated into the objective function. This approach typically involves constructing logarithmic barrier functions that penalize constraint violations, with MATLAB's fmincon function providing built-in support for such implementations through its interior-point algorithm option.

The primal-dual interior point method constitutes a more advanced optimization technique that enhances solving efficiency by simultaneously addressing both primal and dual problems. This method employs predictor-corrector steps and maintains feasibility while approaching optimality, often achieving superior convergence properties compared to pure barrier methods. In MATLAB implementation, this can be realized through customized algorithms using optimoptions to configure optimality tolerances and step sizes, or by developing specialized routines that handle Karush-Kuhn-Tucker (KKT) conditions directly.

These optimization techniques find applications across various practical domains including economics, physics, and engineering optimization problems. For instance, barrier methods can efficiently handle portfolio optimization with inequality constraints, while primal-dual methods excel in large-scale structural optimization problems. To deepen your understanding of MATLAB's optimization capabilities, we recommend consulting MATLAB's official documentation on Optimization Toolbox functions or exploring online tutorials that demonstrate practical implementations with sample code and convergence analysis.