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Linear matrix inequality representation of sets. (English) Zbl 1116.15016
One of the best techniques that are used successfully in many optimization problems is to convert those problems to linear matrix inequalities (LMIs). Unfortunately there is no systematic (or general) way to produce LMIs. Every problem has to be treated by some particular trick. Therefore, the following important question is discussed by the authors: Which types of constraint sets can be converted to LMIs and which do not. They obtain a necessary condition which must hold for a set \(C\subseteq \mathbb R^n\) in order for \(C\) to have an LMI representation. It is proved that the obtained condition is also sufficient for \(n=2\). Several examples are considered to illustrate the obtained results.

15A45 Miscellaneous inequalities involving matrices
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