A Survey of the S-Lemma

A Survey of the S-Lemma

In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. The...

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Bibliographic Details
Published in:SIAM review 2007-09, Vol.49 (3), p.371-418
Main Authors: Pólik, Imre, Terlaky, Tamás
Format: Article
Language:eng
Subjects:
Algebra
Calculus of variations and optimal control
Computational geometry
Control theory
Convex and discrete geometry
Convexity
Ellipsoids
Error analysis
Exact sciences and technology
Geometry
Joints
Linear algebra
Mathematical analysis
Mathematical models
Mathematical theorems
Mathematical vectors
Mathematics
Matrices
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in mathematical programming, optimization and calculus of variations
Optimization
Optimization techniques
Polynomials
Proving
Quadratic functions
Sciences and techniques of general use
Solvability
Studies
Survey and Review
Theory
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Summary:In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry.
ISSN:0036-1445
1095-7200