Applied Optimization. 65. Dordrecht: Kluwer Academic Publishers. xvi, 283 p. EUR 105.00; $ 97.00; £66.00 (2002).
This book is an efficient and well-readable presentation of some selected topics on theory and applications of SDP (semidefinite programming). (SDP) is a special subclass of nonlinear convex optimization which during the last decade became a very active research area in applied mathematics: the number of new results, articles and books on this subject as well as corresponding applications to real-life problems is still growing considerably. Exemplarily, applications in systems, control theory and combinatorial optimization are mentioned here.
The book is divided into four parts: – an introductory Chapter 1, – Part I Theory and Algorithms (Chapters 2-8), – Part II Selected Applications (Chapters 9-13) and - Appendices.
The introductory chapter offers a short overview of the field of (SDP) including a survey on interior point methods as well as applications in combinatorial optimization and engineering.
Part I Theory and Algorithms: Chapter 2: Duality, optimality, and degeneracy Chapter 3: The central path Chapter 4: Self-dual embeddings Chapter 5: The primal logarithmic barrier method Chapter 6: Primal-dual affine-scaling methods Chapter 7: Primal-dual path-following methods Chapter 8: Primal-dual potential reduction methods
Part II Selected Applications Chapter 9: Convex quadratic approximation Chapter 10: The Lovasz function Chapter 11: Graph colouring and the max-K-cut problem Chapter 12: The stability number of a graph Chapter 13: The satisfiability problem.
The appendices contain some background material (on matrix calculus, convex analysis, nonlinear optimization and analytic functions) which is used in this book.
The well-written monograph is self-contained and an updated version of the PhD thesis of the author. It may be useful to several types of readers; for graduate and undergraduate students as well as for mathematicians and engineers dealing with applications that can be modelled by semidefinite calculus.
Altogether, this monograph represents an outstanding contribution to applied mathematics for both students and researchers alike.