Curvature as a Complexity Bound in Interior-Point Methods

In this thesis, we investigate the curvature of interior paths as a component of complexity bounds for interior-point methods (IPMs) in Linear Optimization (LO). LO is an optimization paradigm, where both the objective and the constraints of the model are represented by linear relationships of the decision variables. Among the class ofalgorithms for LO, our focus is on IPMs which have been an extremely active research area in the last three decades. IPMs in optimization are unique in the sense that they enjoy the best iteration-complexity bounds which are polynomial in the size of the LO problem. The main objects of our interest in this thesis are two distinct curvature measures in the literature, the geometric and the Sonnevend curvature of the central path. The central path is a fundamental tool for the design and the study of IPMs and we will see both that the geometric and Sonnevend\u27s curvature of the central path are proven to be useful in approaching the iteration-complexity questions in IPMs. While the Sonnevend curvature of the central path has been rigorously shown to determine the iteration-complexity of certain IPMs, the role of the geometric curvature in the literature to explain the iteration-complexity is still not well-understood. The novel approach in this thesis is to explore whether or not there is a relationship between these two curvature concepts aiming to bring the geometric curvature into the picture. The structure of the thesis is as follows. In the first three chapters, we present the basic knowledge of path-following IPMs algorithms and review the literature on Sonnevend\u27s curvature and the geometric curvature of the central path. In Chapter 4, we analyze a certain class ofLO problems and show that the geometric and Sonnevend\u27s curvature for these problems display analogous behavior. In particular, the main result of this chapter states that in order to establish an upper bound for the total Sonnevend curvature of the central path, it is sufficient to consider only the case when the number of inequalities is twice as big as the dimension. In Chapter 5, we study the redundant Klee-Minty (KM) construction and prove that the classical polynomial upper bound for IPMs is essentially tight for the Mizuno-Todd-Ye predictor-corrector algorithm. This chapter also provides a negative answer to an open problem about the Sonnevend curvature posed by Stoer et al. in 1993. Chapter 6 investigates a condition number relevant to the Sonnevend curvature and yields a strongly polynomial bound for that curvature in some special cases. Chapter 7 deals with another self-concordant barrier function, the volumetric barrier, and the volumetric path. That chapter investigates some of the basic properties of the volumetric path and shows that certain fundamental properties of the central path failto hold for the volumetric path. Chapter 8 concludes the thesis by providing some final remarks and pointing out future research directions