Exact semidefinite programming bounds for packing problems

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $\mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $\pi/3$ on the hemisphere in $\mathbb R^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (2\sqrt{2}-1)/7$, is sharp by rounding to $\mathbb Q[\sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.Comment: 24 page

Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programming

A barrier certificate often serves as an inductive invariant that isolates an unsafe region from the reachable set of states, and hence is widely used in proving safety of hybrid systems possibly over an infinite time horizon. We present a novel condition on barrier certificates, termed the invariant barrier-certificate condition, that witnesses unbounded-time safety of differential dynamical systems. The proposed condition is the weakest possible one to attain inductive invariance. We show that discharging the invariant barrier-certificate condition -- thereby synthesizing invariant barrier certificates -- can be encoded as solving an optimization problem subject to bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm based on difference-of-convex programming, which approaches a local optimum of the BMI problem via solving a series of convex optimization problems. This algorithm is incorporated in a branch-and-bound framework that searches for the global optimum in a divide-and-conquer fashion. We present a weak completeness result of our method, namely, a barrier certificate is guaranteed to be found (under some mild assumptions) whenever there exists an inductive invariant (in the form of a given template) that suffices to certify safety of the system. Experimental results on benchmarks demonstrate the effectiveness and efficiency of our approach.Comment: To be published in Inf. Comput. arXiv admin note: substantial text overlap with arXiv:2105.1431

Exact algorithms for semidefinite programs with degenerate feasible set

International audienceLet A 0 ,. .. , A n be m × m symmetric matrices with entries in Q, and let A(x) be the linear pencil A 0 + x 1 A 1 + · · · + x n A n , where x = (x 1 ,. .. , x n) are unknowns. The linear matrix inequality (LMI) A(x) 0 defines the subset of R n , called spectrahedron, containing all points x such that A(x) has non-negative eigenvalues. The minimization of linear functions over spectrahedra is called semidefinite programming (SDP). 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 solving SDP are mostly based on the interior point method, assuming some non-degeneracy properties such as the existence of interior points in the admissible set. 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 but we prove that solving such problems can be done in polynomial time if either n or m is fixed

Exact algorithms for semidefinite programs with degenerate feasible set

International audienceGiven symmetric matricesA0, A1, . . . , Anof sizemwith rational entries, theset of real vectorsx= (x1, . . . , xn) such that the matrixA0+x1A1+···+xnAnhas non-negative eigenvalues is called a spectrahedron. Minimizationof linear functions over spectrahedra is called semidefinite programming.Such problems appear frequently in control theory and real algebra, espe-cially in the context of nonnegativity certificates for multivariate polynomi-als based on sums of squares.Numerical software for semidefinite programming are mostlybased oninterior point methods, assuming non-degeneracy properties such as the ex-istence of an interior point in the spectrahedron. In this paper, we designan exact algorithm based on symbolic homotopy for solving semidefiniteprograms without assumptions on the feasible set, and we analyze its com-plexity. Because of the exactness of the output, it cannot compete withnumerical routines in practice. However, we prove that solving such prob-lems can be done in polynomial time if eithernormis fixed