Journal
SIAM REVIEW
Volume 49, Issue 3, Pages 371-418Publisher
SIAM PUBLICATIONS
DOI: 10.1137/S003614450444614X
Keywords
S-lemma; S-procedure; control theory; nonconvex theorem of alternatives; numerical range; relaxation theory; semidefinite optimization; generalized convexities
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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.
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