Let @math be a linear matrix, or pencil, generated by given symmetric matrices @math of size @math with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semi-algebraic set called spectrahedron, described by a linear matrix inequality (LMI). 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 @math 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 B 'ezout bound of @math . When @math (resp. @math ) is fixed, such a bound, and hence the complexity, is polynomial in @math (resp. @math ). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms.
On input @math , @math can be defined by @math polynomial inequalities in @math of degree @math (see e.g. @cite_34 ). As far as we know, the state-of-the-art for designing algorithms deciding the emptiness of @math consists only of algorithms for deciding the emptiness of general semi-algebraic sets; our contribution being the first attempt to exploit structural properties of the problem, e.g. through the smallest rank property (Theorem ).
@cite_34
2003232014
Abstract We present an algorithm to determine if a real polynomial is a sum of squares (of polynomials), and to find an explicit representation if it is a sum of squares. This algorithm uses the fact that a sum of squares representation of a real polynomial corresponds to a real, symmetric, positive semi-definite matrix whose entries satisfy certain linear equations.