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EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES.

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Title:	EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES.
Authors:	HENRION, DIDIER¹ henrion@laas.fr, NALDI, SIMONE² caponord2007@gmail.com, EL DIN, MOHAB SAFEY³ mohab.safey@lip6.fr
Source:	SIAM Journal on Optimization. 2016, Vol. 26 Issue 4, p2512-2539. 28p.
Subjects:	Linear matrix inequalities, Algorithms, Polynomials, Quadratic programming, Topological degree
Abstract:	Let A(x) = A0 + x1A1 +...+xnAn be a linear matrix, or pencil, generated by given symmetric matrices A0,A1,...,An of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bézout bound of d. When m (resp., n) is fixed, such a bound, and hence the complexity, is polynomial in n (resp., m). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

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Abstract:	Let A(x) = A0 + x1A1 +...+xnAn be a linear matrix, or pencil, generated by given symmetric matrices A0,A1,...,An of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bézout bound of d. When m (resp., n) is fixed, such a bound, and hence the complexity, is polynomial in n (resp., m). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms. [ABSTRACT FROM AUTHOR]
ISSN:	10526234
DOI:	10.1137/15M1036543