Synopses & Reviews
The book provides an overview of the research in interior point methods since the publication of N. Karmarkar's seminal paper in 1984. Leading international experts have contributed to the book with summaries of their relevant areas of specialization. Part I gives an overview of basic variants of interior point algorithms for linear programming. Duality-theory for LP and sensitivity analysis is developed. Results on the affine scale, path following, potential reduction, infeasible interior point methods and implementation strategies are discussed. Part II deals with nonlinear programming. Here necessary smoothness conditions are introduced and illustrated. Algorithms for general smooth convex problems, complementarity and semidefinite optimization problems are presented. The implementation of barrier methods for general nonlinear optimization problems is also considered. Part III covers some application areas such as combinatorial optimization, global optimization and VLSI design. Audience: Researchers and graduate students in mathematical programming, operations research, management science, combinatorics and industrial engineering; also professionals in industry who work on problems using linear programming.
Synopsis
One has to make everything as simple as possible but, never more simple. Albert Einstein Discovery consists of seeing what every- body has seen and thinking what nobody has thought. Albert S. ent_Gyorgy; The primary goal of this book is to provide an introduction to the theory of Interior Point Methods (IPMs) in Mathematical Programming. At the same time, we try to present a quick overview of the impact of extensions of IPMs on smooth nonlinear optimization and to demonstrate the potential of IPMs for solving difficult practical problems. The Simplex Method has dominated the theory and practice of mathematical pro- gramming since 1947 when Dantzig discovered it. In the fifties and sixties several attempts were made to develop alternative solution methods. At that time the prin- cipal base of interior point methods was also developed, for example in the work of Frisch (1955), Caroll (1961), Huard (1967), Fiacco and McCormick (1968) and Dikin (1967). In 1972 Klee and Minty made explicit that in the worst case some variants of the simplex method may require an exponential amount of work to solve Linear Programming (LP) problems. This was at the time when complexity theory became a topic of great interest. People started to classify mathematical programming prob- lems as efficiently (in polynomial time) solvable and as difficult (NP-hard) problems. For a while it remained open whether LP was solvable in polynomial time or not. The break-through resolution ofthis problem was obtained by Khachijan (1989).
Table of Contents
Preface.
Part I: Linear Programming. 1. Introduction to the Theory of Interior Point Methods;
B. Jansen, et al. 2. Affine Scaling Algorithm;
T. Tsuchiya. 3. Target-Following Methods for Linear Programming;
B. Jansen, et al. 4. Potential Reduction Algorithms;
K.M. Anstreicher. 5. Infeasible-Interior-Point Algorithms;
S. Mizuno. 6. Implementation of Interior-Point Methods for Large Scale Linear Programs;
E.D. Andersen, et al. Part II: Convex Programming. 7. Interior-Point Methods for Classes of Convex Programs;
F. Jarre. 8. Complementarity Problems;
A. Yoshise. 9. Semidefinite Programming;
M.V. Ramana, P.M. Pardalos. 10. Implementing Barrier Methods for Nonlinear Programming;
D.F. Shanno, et al. Part III: Applications, Extensions. 11. Interior Point Methods for Combinatorial Optimization;
J.E. Mitchell. 12. Interior Point Methods for Global Optimization;
P.M. Pardalos, M.G.C. Resende. 13. Interior Point Approaches for the VLSI Placement Problem;
A. Vannelli, et al.