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Equivalence of sum of squares convex relaxations for quadratic distance problems
Author(s) -
Garulli Andrea,
Masi Alfio,
Vicino Antonio
Publication year - 2013
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.2810
Subject(s) - explained sum of squares , mathematics , equivalence (formal languages) , quadratic equation , regular polygon , relaxation (psychology) , least squares function approximation , convex optimization , representation (politics) , polynomial , discrete mathematics , mathematical optimization , combinatorics , mathematical analysis , statistics , geometry , estimator , politics , political science , law , psychology , social psychology
SUMMARY This paper deals with convex relaxations for quadratic distance problems, a class of optimization problems relevant to several important topics in the analysis and synthesis of robust control systems. Some classes of convex relaxations are investigated using the sum of squares paradigm for the representation of positive polynomials. The main contribution is to show that two different relaxations, based respectively on the Positivstellensatz and on properties of homogeneous polynomial forms, are equivalent. Relationships among the considered relaxations are discussed and numerical comparisons are presented, highlighting their degree of conservatism. Copyright © 2012 John Wiley & Sons, Ltd.