Interior Point Methods for Nonlinear Optimization

Abstract

Interior-point methods (IPMs) are among the most efficient methods for solving linear, and also wide classes of other convex optimization problems. Since the path-breaking work of Karmarkar [48], much research was invested in IPMs. Many algorithmic variants were developed for Linear Optimization (LO). The new approach forced to reconsider all aspects of optimization problems. Not only the research on algorithms and complexity issues, but implementation strategies, duality theory and research on sensitivity analysis got also a new impulse. After more than a decade of turbulent research, the IPM community reached a good understanding of the basics of IPMs. Several books were published that summarize and explore different aspects of IPMs. The seminal work of Nesterov and Nemirovski [63] provides the most general framework for polynomial IPMs for convex optimization. Den Hertog [42] gives a thorough survey of primal and dual path-following IPMs for linear and structured convex optimization problems. Jansen [45] discusses primal-dual target following algorithms for linear optimization and complementarity problems.Wright [93] also concentrates on primal-dual IPMs, with special attention on infeasible IPMs, numerical issues and local, asymptotic convergence properties. The volume [80] contains 13 survey papers that cover almost all aspects of IPMs, their extensions and some applications. The book of Ye [96] is a rich source of polynomial IPMs not only for LO, but for convex optimization problems as well. It extends the IPM theory to derive bounds and approximations for classes of nonconvex optimization problems as well. Finally, Roos, Terlaky and Vial [72] present a thorough treatment of the IPM based theory - duality, complexity, sensitivity analysis - and wide classes of IPMs for LO.