Advances in Newton-based Barrier Methods for Nonlinear Programming
Nonlinear programming is a very important tool for optimizing many systems in science and engineering. The interior point solver IPOPT has become one of the most popular solvers for NLP because of its high performance. However, certain types of problems are still challenging for IPOPT. This dissertation considers three improvements or extensions to IPOPT to improve performance on several practical classes of problems. Compared to active set solvers that treat inequalities by identifying active constraints and transforming to equalities, the interior point method is less robust in the presence of degenerate constraints. Interior point methods require certain regularity conditions on the constraint set for the solution path to exist. Dependent constraints commonly appear in applications such as chemical process models and violate the regularity conditions. The interior point solver IPOPT introduces regularization terms to attempt to correct this, but in some cases the required regularization terms either too large or too small and the solver will fail. To deal with these challenges, we present a new structured regularization algorithm, which is able to numerically delete dependent equalities in the KKT matrix. Numerical experiments on hundreds of modified example problems show the effectiveness of this approach with average reduction of more than 50% of the iterations. In some contexts such as online optimization, very fast solutions of an NLP are very important. To improve the performance of IPOPT, it is best to take advantage of problem structure. Dynamic optimization problems are often called online in a control or stateestimation. These problems are very large and have a particular sparse structure. This work investigates the use of parallelization to speed up the NLP solution. Because the KKT factorization is the most expensive step in IPOPT, this is the most important step to parallelize. Several cyclic reduction algorithms are compared for their performance on generic test matrices as well as matrices of the form found in dynamic optimization. The results show that for very large problems, the KKT matrix factorization time can be improved by a factor of four when using eight processors. Mathematical programs with complementarity constraints (MPCCs) are another challenging class of problems for IPOPT. Several algorithmic modifications are examined to specially handle the difficult complementarity constraints. First, two automatic penalty adjustment approaches are implemented and compared. Next, the use of our structured regularization is tested in combination with the equality reformulation of MPCCs. Then, we propose an altered equality reformulation of MPCCs which effectively removes the degenerate equality or inequality constraints. Using the MacMPEC test library and two applications, we compare the efficiency of our approaches to previous NLP reformulation strategies.