Search

In this chapter, we discuss a modification of the standard path-following scheme that tends to speed up the global convergence. This modification, the central region method, generates iterates that do not really trace the central path, or at least not closely. In this way, it has a relatively large freedom of movement, and consequently the ability to take long steps. This makes it interesting to consider more sophisticated search directions. We propose a search direction that is built up in three phases, viz. 1. Initial centering, 2. Predictor, 3. Second order centrality corrector.

Modern solvers for linear and quadratic programming often incorporate some variant of the primal-dual path-following method, which is a class of interior point methods. This method turns out to be suited for solving semidefinite programming problems as well. In this chapter, we propose a framework for analyzing primal-dual path-following algorithms for semidefinite programming. Our framework is a direct extension of the v-space approach, which was developed by Kojima et al. [63] in the context of complementarity problems.

List of Figures. List of Tables. Preface. Contributing Authors. Part I: Theory and Algorithms of Semidefinite Programming J.F. Sturm. 1. Introduction. 2. Duality. 3. Polynomiality of Path-following Methods. 4. Self-Dual Embedding Technique. 5. Properties of the Central Path. 6. Superlinear Convergence. 7. Central Region Methods. Part II: Linear, Quadratic, Semidefinite Programming and Beyond. 8. An Implementation of the Homogeneous Algorithm E.D. Andersen, K.D. Andersen. 9. A Simplified Correctness Proof for Interior Point Algorithm S.A. Vavasis, Y. Ye. 10. New Analysis of Newton Methods for LCP J. Peng, et al. 11. Numerical Evaluation of SDPA K. Fujisawa, et al. 12. Robust Modeling of Multi-Stage Portfolio Problems A. Ben-Tal, et al. 13. An Interior Point SQP Parallel B&B Method. 14. Solving Linear Ordering Problems J.E. Mitchell, B. Borchers. 15. Finite Element Methods for Solving Parabolic Inverse Problems Y.L. Keung, J. Zou. 16. Error Bounds For Quadratic Systems Z.-Q. Luo, J.F. Sturm. 17. Squared Functional Systems and Optimization Problems Y. Nesterov. 18. Interior Point Methods: Current Status and Future Directions R.M. Freund, S. Mizuno. Index.