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

Barrier methods and primal-dual interior-point methods are effective numerical approaches for solving inequality-constrained convex optimization problems, which can be efficiently implemented using MATLAB algorithms. The barrier method transforms inequality-constrained problems into a sequence of unconstrained optimization problems by adding logarithmic barrier terms to the objective function. As the barrier parameter decreases, the solution progressively approximates the optimal solution of the original problem. Key implementation considerations include selecting appropriate barrier parameter update strategies and initial points to ensure rapid algorithm convergence. In MATLAB, this typically involves creating a custom objective function with barrier terms and implementing a parameter reduction scheme using while-loops or for-loops. Primal-dual interior-point methods simultaneously optimize primal and dual variables by introducing slack variables to convert inequality constraints into equality constraints, then solving them iteratively using Newton's method. Compared to barrier methods, primal-dual methods generally exhibit faster convergence rates but require more complex implementation. MATLAB implementation involves solving linear equation systems at each iteration using built-in solvers like backslash operator (\) or linsolve(), while maintaining strict feasibility conditions through careful step-size selection. When implementing these methods in MATLAB, developers can either utilize the built-in Optimization Toolbox or write custom iterative solving code. For barrier methods, programmers need to construct objective functions containing barrier terms and gradually adjust barrier parameters through systematic reduction protocols. Primal-dual interior-point methods require solving sequences of linear equation systems while ensuring variables maintain strict feasibility conditions throughout iterations, often implemented using predictor-corrector algorithms. Both methods have distinct advantages: barrier methods offer easier implementation but slower convergence, while primal-dual interior-point methods provide faster convergence at the cost of greater sensitivity to initial point selection. Practical applications should choose the appropriate method based on problem scale and performance requirements, with barrier methods suitable for simpler problems and primal-dual methods preferred for large-scale optimization where convergence speed is critical.