Non-interior path-following methods for complementarity problems

Abstract

Because of its excellent numerical performance, non-interior path following methods (also called smoothing methods) have become an important class of methods for solving complementarity problems. However, no rate of convergence results are available for these methods. In this thesis, we bridge this gap between the theory and the practical performance of the methods. Specifically, we focus on the rates of convergence, the complexity, and the implementation of non-interior path following methods.The thesis introduces new notions of neighborhoods of the central path for non-interior path following methods for linear complementarity problems. These neighborhoods are modeled on similar concepts from the interior point literature and are used to adjust the value of a continuation parameter. However, these neighborhoods are fundamentally different from those used in the interior-point methods. In particular the solution set of the underlying LCP is contained in the interior of these neighborhoods relative to the affine constraints. The new neighborhood concepts have proven to be fundamental for both the theoretical analysis of the algorithms and in their practical implementation. With these new neighborhood concepts, we are able to establish the first global linear convergence result for non-interior path following methods. In order to accelerate the convergence, we introduce a predictor-corrector strategy. This strategy allows us to construct the first predictor-corrector non-interior path following method that is both globally linearly convergent and locally quadratically convergent. In the thesis, we also make progress toward understanding the computational complexity of these methods. Complexity results are obtained from both the algorithmic and condition-based perspectives. The complexity bounds that we establish are the only results for these methods that are available to date. These results represent a first step toward understanding the complexity of non-interior path following methods.