Testing the Primal-Dual Interior Point Method for Nonlinear Optimization Problems

The Primal-Dual Interior Point Method is an efficient numerical approach for solving nonlinear optimization problems, particularly effective in constrained optimization scenarios. This method introduces slack variables and dual variables to transform the original problem into a series of unconstrained subproblems, utilizing Newton iterations to progressively approach the optimal solution. In code implementation, this involves formulating the KKT conditions and solving the linearized system using techniques like LU decomposition or iterative solvers.

In the testing program, the algorithm's core lies in appropriately setting the barrier parameter and step size adjustment strategy to ensure rapid convergence while maintaining constraint feasibility. By selecting a suitable initial point and employing adaptive adjustment mechanisms, the program can effectively avoid local optima and numerical instability. In practice, this requires implementing safeguards like fraction-to-boundary rules and merit function evaluations to guide the iteration process.

Convergence validation typically involves two aspects: verifying whether the objective function value follows the expected theoretical descent trend, and monitoring if the duality gap approaches zero as iterations increase. Efficient implementations often incorporate line search techniques to accelerate convergence, alongside matrix factorization methods (e.g., sparse Cholesky decomposition) to address computational bottlenecks in large-scale problems. Code-wise, this entails implementing backtracking line search and exploiting problem structure through specialized linear algebra routines.

Test results demonstrate that with proper parameter settings, the method typically achieves machine-precision optimal solutions within tens of iterations for medium-scale nonlinear optimization problems, showing particular effectiveness for convex optimization problems with smooth objective functions and constraints. The implementation can be optimized further by using analytical Hessians where available and implementing warm-start strategies for related problem sequences.