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.
All the above algorithms do not exploit the particular structure of spectrahedra understood as determinantal semi-algebraic sets. In @cite_30 , the authors showed that deciding emptiness of @math can be done in time @math , that is in polynomial time in @math (resp. linear time in @math ) if @math (resp. @math ) is fixed . The main drawback of this algorithm is that it is based on general procedures for quantifier elimination, and hence it does not lead to efficient practical implementations. Note also that the complexity constant in the exponent is still unknown.
@cite_30
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We show that the feasibility of a system of m linear inequalities over the cone of symmetric positive semidefinite matrices of order n can be tested in mn^O(min m,n^2 ) arithmetic operations with ln^O(min m,n^2 )-bit numbers, where l is the maximum binary size of the input coefficients. We also show that any feasible system of dimension (m,n) has a solution X such that log||X|| ≤ ln^O(min m,n^2 ).