Yurii Nesterov and Arkadii Nemirovskii
Studies in Applied Mathematics 13
Written for specialists working in optimization, mathematical programming, or control theory. The general theory of pathfollowing and potential reduction interior point polynomial time methods, interior point methods, interior point methods for linear and quadratic programming, polynomial time methods for nonlinear convex programming, efficient computation methods for control problems and variational inequalities, and acceleration of pathfollowing methods are covered.
In this book, the authors describe the first unified theory of polynomialtime interiorpoint methods. Their approach provides a simple and elegant framework in which all known polynomialtime interiorpoint methods can be explained and analyzed; this approach yields polynomialtime interiorpoint methods for a wide variety of problems beyond the traditional linear and quadratic programs.
The book contains new and important results in the general theory of convex programming, e.g., their "conic" problem formulation in which duality theory is completely symmetric. For each algorithm described, the authors carefully derive precise bounds on the computational effort required to solve a given family of problems to a given precision. In several cases they obtain better problem complexity estimates than were previously known. Several of the new algorithms described in this book, e.g., the projective method, have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.
Audience
Specialists working in the areas of optimization, mathematical programming, or control theory will find this book invaluable for studying interiorpoint methods for linear and quadratic programming, polynomialtime methods for nonlinear convex programming, and efficient computational methods for control problems and variational inequalities. A background in linear algebra and mathematical programming is necessary to understand the book. The detailed proofs and lack of "numerical examples" might suggest that the book is of limited value to the reader interested in the practical aspects of convex optimization, but nothing could be further from the truth. An entire chapter is devoted to potential reduction methods precisely because of their great efficiency in practice.
Contents
Chapter 1: SelfConcordant Functions and Newton Method; Chapter 2: PathFollowing InteriorPoint Methods; Chapter 3: Potential Reduction InteriorPoint Methods; Chapter 4: How to Construct Self Concordant Barriers; Chapter 5: Applications in Convex Optimization; Chapter 6: Variational Inequalities with Monotone Operators; Chapter 7: Acceleration for Linear and Linearly Constrained Quadratic Problems; Bibliography; Appendix 1; Appendix 2.
1994 / ix + 405 pages / Softcover / ISBN13: 9780898715156 / ISBN10: 0898715156 / List Price $125.50 / SIAM Member Price $87.85 / Order Code AM13
