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Semidefinite programming (SDP) is one of the most exciting and active research areas in optimization. It has and continues to attract researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity has been prompted by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interior-point algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory.
The Handbook of Semidefinite Programming offers an advanced and broad overview of the current state of the field. It contains nineteen chapters written by the leading experts on the subject. The chapters are organized in three parts: Theory, Algorithms, and Applications and Extensions.
List of contents
1 Introduction.- I Theory.- 2 Convex Analysis on Symmetric Matrices.- 3 The Geometry of Semidefinite Programming.- 4 Duality and Optimality Conditions.- 5 Self-Dual Embeddings.- 6 Robustness 139.- 7 Error Analysis 163.- II Algorithms.- 8 Symmetric Cones, Potential Reduction Methods.- 9 Potential Reduction and Primal-Dual Methods.- 10 Path-Following Methods.- 11 Bundle Methods and Eigenvalue Functions.- III Applications and Extensions.- 12 Combinatorial Optimization.- 13 Nonconvex Quadratic Optimization.- 14 SDP in Systems and Control Theory.- 15 Structural Design.- 16 Moment Problems and Semidefinite Optimization.- 17 Design of Experiments in Statistics.- 18 Matrix Completion Problems.- 19 Eigenvalue Problems and Nonconvex Minimization.- 20 General Nonlinear Programming.- References.- A-.1 Conclusion and Further Historical Notes.- A-.1.1 Combinatorial Problems.- A-.l.2 Complementarity Problems.- A-.l.3 Complexity, Distance to III-Posedness, and Condition Numbers.- A-.1.4 Cone Programming.- A-.1.5 Eigenvalue Functions.- A-.1.6 Engineering Applications.- A-.1.7 Financial Applications.- A-.1.8 Generalized Convexity.- A-.1.9 Geometry.- A-.l.10 Implementation.- A-.1.11 Matrix Completion Problems.- A-.1.12 Nonlinear and Nonconvex SDPs.- A-.1.13 Nonlinear Programming.- A-.1.14 Quadratic Constrained Quadratic Programs.- A-.1.15 Sensitivity Analysis.- A-. 1.16 Statistics.- A-. 1.17 Books and Related Material.- A-.1.18 Review Articles.- A-.1.19 Computer Packages and Test Problems.- A-.2 Index.
Summary
This tremendous research activity has been prompted by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interior-point algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory.
