Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field K. For K-polyhedra and linear programming over K we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral K-spectrahedra, as well as sets that are semidefinite-representable over K but are not K-spectrahedra.

We consider linear matrix inequalities (LMIs) A = A0 + x1A1 + ⋅⋅⋅ + xnAn⪰0 with the Ai’s being m × m symmetric matrices, with entries in a ring \(\mathcal {R}\). When \(\mathcal {R}= \mathbb {R}\), the feasibility problem consists in deciding whether the xi’s can be instantiated to obtain a positive semi-definite matrix. When \(\mathcal {R}= \mathbb {Q}[y_1, \ldots , y_t]\), the problem asks for a formula on the parameters y1, …, yt, which describes the values of the parameters for which the specialized LMI is feasible. This problem can be solved using general quantifier elimination algorithms, with a complexity that is exponential in n. In this work, we leverage the LMI structure of the problem to design an algorithm that computes a formula Φ describing a dense subset of the feasible region of parameters, under genericity assumptions. The complexity of this algorithm is exponential in n, m and t but becomes polynomial in n when m and t are fixed. We apply the algorithm to a parametric sum-of-squares problem and to the convergence analyses of certain first-order optimization methods, which are both known to be equivalent to the feasibility of certain parametric LMIs, hence demonstrating its practical interest.

Given symmetric matrices A0,A1,…,An of size m with rational entries, the set of real vectors x=(x1,…,xn) such that the matrix A0+x1A1+⋯+xnAn has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice. However, we prove that solving such problems can be done in polynomial time if either n or m is fixed.

Let f1, ..., fm be elements in a quotient Rn/N which has finite dimension as a K-vector space, where R = K[X1, ..., Xr] and N is an R-submodule of Rn. We address the problem of computing a Gröbner basis of the module of syzygies of (f1, ..., fm), that is, of vectors (p1, ..., pm) ∈ Rm such that p1f1 + ... + pm fm = 0.

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.

Let H0, …, H n be m x m matrices with entries in Q and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix H(x) = H0+x1H_1+…+xnHn and the problem of computing sample points in each connected component of the real algebraic set defined by the rank constraint rank}(H(x))≤ r, for a given integer r ≤ m-1. Computing sample points in real algebraic sets defined by rank defects in linear matrices is a general problem that finds applications in many areas such as control theory, computational geometry, optimization, etc. Moreover, Hankel matrices appear in many areas of engineering sciences. Also, since Hankel matrices are symmetric, any algorithmic development for this problem can be seen as a first step towards a dedicated exact algorithm for solving semi-definite programming problems, i.e. linear matrix inequalities. Under some genericity assumptions on the input (such as smoothness of an incidence variety), we design a probabilistic algorithm for tackling this problem. It is an adaptation of the so-called critical point method that takes advantage of the special structure of the problem. Its complexity reflects this: it is essentially quadratic in specific degree bounds on an incidence variety. We report on practical experiments and analyze how the algorithm takes advantage of this special structure. A first implementation outperforms existing implementations for computing sample points in general real algebraic sets: it tackles examples that are out of reach of the state-of-the-art.