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Linear matrix inequality representation of sets
Author(s) -
Helton J. William,
Vinnikov Victor
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20155
Subject(s) - convexity , mathematics , representation (politics) , conjecture , set (abstract data type) , matrix (chemical analysis) , perspective (graphical) , scope (computer science) , order (exchange) , mathematical economics , algebra over a field , pure mathematics , computer science , law , materials science , politics , political science , composite material , geometry , finance , financial economics , economics , programming language
This article concerns the question, Which subsets of ℝ m can be represented with linear matrix inequalities (LMIs)? This gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory. Also, before having much hope of representing engineering problems as LMIs by automatic methods, one needs a good idea of which problems can and cannot be represented by LMIs. Little is currently known about such problems. In this article we give a necessary condition that we call “rigid convexity,” which must hold for a set ⊆ ℝ m in order for to have an LMI representation. Rigid convexity is proved to be necessary and sufficient when m = 2. This settles a question formally stated by Pablo Parrilo and Berndt Sturmfels in [15]. As shown by Lewis, Parillo, and Ramana [11], our main result also establishes (in the case of three variables) a 1958 conjecture by Peter Lax on hyperbolic polynomials. © 2006 Wiley Periodicals, Inc.