Solving rank-constrained semidefinite programs in exact arithmetic

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix is fixed, the complexity is polynomial in the number of variables. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of Symbolic Computatio

Elimination for generic sparse polynomial systems

We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an l-dimensional coordinate affine space with l < n. The complexity of the algorithm depends polynomially on combinatorial invariants associated to the supports.Comment: 22 page