Convex pencils of real quadratic forms

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S^n; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)\leq 2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number b_k(X)\leq 2(k+2).Comment: Updated version to be published in DC

Semidefinite geometry of the numerical range

The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite cone. Both primal and dual sets can also be viewed as convex hulls of explicit algebraic plane curve components. Several numerical examples illustrate this interplay between algebra, geometry and semidefinite programming duality. Finally, these techniques are used to revisit a theorem in statistics on the independence of quadratic forms in a normally distributed vector

A Semidefinite Hierarchy for Containment of Spectrahedra

A spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program. We propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one. This approach comes from applying a moment relaxation to a suitable polynomial optimization formulation. The hierarchical criterion is stronger than a solitary semidefinite criterion discussed earlier by Helton, Klep, and McCullough as well as by the authors. Moreover, several exactness results for the solitary criterion can be brought forward to the hierarchical approach. The hierarchy also applies to the (equivalent) question of checking whether a map between matrix (sub-)spaces is positive. In this context, the solitary criterion checks whether the map is completely positive, and thus our results provide a hierarchy between positivity and complete positivity.Comment: 24 pages, 2 figures; minor corrections; to appear in SIAM J. Opti

A Revisit to Quadratic Programming with One Inequality Quadratic Constraint via Matrix Pencil

The quadratic programming over one inequality quadratic constraint (QP1QC) is a very special case of quadratically constrained quadratic programming (QCQP) and attracted much attention since early 1990's. It is now understood that, under the primal Slater condition, (QP1QC) has a tight SDP relaxation (PSDP). The optimal solution to (QP1QC), if exists, can be obtained by a matrix rank one decomposition of the optimal matrix X? to (PSDP). In this paper, we pay a revisit to (QP1QC) by analyzing the associated matrix pencil of two symmetric real matrices A and B, the former matrix of which defines the quadratic term of the objective function whereas the latter for the constraint. We focus on the \undesired" (QP1QC) problems which are often ignored in typical literature: either there exists no Slater point, or (QP1QC) is unbounded below, or (QP1QC) is bounded below but unattainable. Our analysis is conducted with the help of the matrix pencil, not only for checking whether the undesired cases do happen, but also for an alternative way to compute the optimal solution in comparison with the usual SDP/rank-one-decomposition procedure.Comment: 22 pages, 0 figure

Open problems, questions, and challenges in finite-dimensional integrable systems

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version

Semidefinite geometry of the numerical range

The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite cone. Both primal and dual sets can also be viewed as convex hulls of explicit algebraic plane curve components. Several numerical examples illustrate this interplay between algebra, geometry and semidefinite programming duality. Finally, these techniques are used to revisit a theorem in statistics on the independence of quadratic forms in a normally distributed vector

Polynomials with and without determinantal representations

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal representation. Br\"and\'en has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. So the generalized Lax conjecture fails badly. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation, improving upon Br\"and\'en's mostly unconstructive result. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence