Performance Optimization

Abstract

For a long time the techniques of solving linear optimization (LP) problems improved only marginally. Fifteen years ago, however, a revolutionary discovery changed everything. A new `golden age' for optimization started, which is continuing up to the current time. What is the cause of the excitement? Techniques of linear programming formed previously an isolated body of knowledge. Then suddenly a tunnel was built linking it with a rich and promising land, part of which was already cultivated, part of which was completely unexplored. These revolutionary new techniques are now applied to solve conic linear problems. This makes it possible to model and solve large classes of essentially nonlinear optimization problems as efficiently as LP problems. This volume gives an overview of the latest developments of such `High Performance Optimization Techniques'. The first part is a thorough treatment of interior point methods for semidefinite programming problems. The second part reviews today's most exciting research topics and results in the area of convex optimization. Audience: This volume is for graduate students and researchers who are interested in modern optimization techniques. 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 Method -- 8 An Implementation of the Homogeneous Algorithm -- 9 A Simplified Correctness Proof for Interior Point Algorithm -- 10 New Analysis of Newton Methods for LCP -- 11 Numerical Evaluation of SDPA -- 12 Robust Modeling of Multi-Stage Portfolio Problems -- 13 An Interior Point SQP Parallel B & B Method -- 14 Solving Linear Ordering Problems -- 15 Finite Element Methods for Solving Parabolic Inverse Problems -- 16 Error Bounds For Quadratic Systems -- 17 Squared Functional Systems and Optimization Problems -- 18 Interior Point Methods: Current Status and Future Directions.