Tight Sum-of-Squares lower bounds for binary polynomial optimization problems

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree $2d$ and an odd number of variables $n$, we prove that $\frac{n+2d-1}{2}$ levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires $n$ levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is $n-1$. We disprove this conjecture and derive lower and upper bounds for the rank

New and improved bounds on the contextuality degree of multi-qubit configurations

We present algorithms and a C code to decide quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022), but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from two to seven. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension two and higher, (ii) non-existence of negative subspaces of dimension three and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for ranks four, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree.Comment: 21 pages, 5 figures, 2 tables, submitte

Registration techniques for computer assisted orthopaedic surgery

The registration of 3D preoperative medical data to patients is a key task in developing computer assisted surgery systems. In computer assisted surgery, the patient in the operation theatre must be aligned with the coordinate system in which the preoperative data has been acquired, so that the planned surgery based on the preoperative data can be carried out under the guidance of the computer assisted surgery system.The aim of this research is to investigate registration algorithms for developing computer assisted bone surgery systems. We start with reference mark registration. New interpretations are given to the development of well knowm algorithms based on singular value decomposition, polar decomposition techniques and the unit quaternion representation of the rotation matrix. In addition, a new algorithm is developed based on the estimate of the rotation axis. For non-land mark registration, we first develop iterative closest line segment and iterative closest triangle patch registrations, similar to the well known iterative closest point registration, when the preoperative data are dense enough. We then move to the situation where the preoperative data are not dense enough. Implicit fitting is considered to interpolate the gaps between the data . A new ellipsoid fitting algorithm and a new constructive implicit fitting strategy are developed. Finally, a region to region matching procedure is proposed based on our novel constructive implicit fitting technique. Experiments demonstrate that the new algorithm is very stable and very efficient

Lower bounds on the size of semidefinite programming relaxations

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than $2^{n^c}$, for some constant $c > 0$. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-$O(1)$ sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for MAX-3-SAT

Dualities in Convex Algebraic Geometry

Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre's representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality.Comment: 48 pages, 11 figure

Self-Motions of 3-RPS Manipulators

International audienceRecently a complete kinematic description of the 3-RPS parallel manipulator was obtained using algebraic constraint equations. It turned out that the workspace splits into two components describing two kinematically different operation modes. In this paper the algebraic description is used to give a complete analysis of all possible self-motions of this manipulator in both operation modes. Furthermore it is shown that a transition from one operation mode into the other in a self-motion is possible