The primary goal of this monograph is to introduce a new framework for the theory of primal-dual interior-point methods (IPMs) based on the notion of the self-regularity of a function. Starting from Karmarkar's epoch-making paper [52] in 1984, the research on IPMs has dominated the field of continuous optimization for more than 15 years, and over 3000 papers have been published relating to IPMs. With the efforts of many excellent experts in the field, particularly the path-breaking work by Nesterov and Nemirovskii [83], the theory of IPMs has been developed into a rather mature principle.
An important concept in the IPM literature is the central path of the underlying problem. This was first recognized by Sonnevend [102] and Megiddo [70]. The primal-dual framework presented in [70] laid down the bedrock of many practically efficient IPMs with interesting theoretical properties. These IPMs utilize various strategies to follow the central path approximately. Two contrasting approaches in the analysis and implementation of IPMs are the socalled small-update (with small neighborhood) and large-update (with large neighborhood) methods. Unfortunately, there is still a big gap between the theory and the practical performance of IPMs with respect to these two strategies. As stated by Renegar [97, pp. 51], “It is one of the ironies of the IPM literature that algorithms which are more efficient in practice often have somewhat worse complexity bounds.”
The motivation for this work is to bridge this gap. We deal with linear optimization, nonlinear complementarity problems, semidefinite optimization and second-order conic optimization problems. Our framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered may be interpreted as a path-following method or a constrained potential reduction method. Starting from a primal-dual strictly feasible point, our algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By exploring extensively some intriguing properties of the self-regular function, we establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs.
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