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

The matricial relaxation of a linear matrix inequality

Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask: (Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables x_j. With this relaxation, the domination questions (Q1) and (Q2) have elegant answers, indeed reduce to constructible semidefinite programs. Assume there is an X such that L_1(X) and L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^* L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are unitarily equivalent. Such algebraic certificates are typically called Positivstellensaetze and the above are examples of such for linear polynomials. The paper goes on to derive a cleaner and more powerful Putinar-type Positivstellensatz for polynomials positive on a bounded set of the form {X | L(X) PsD}. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map from a subspace of matrices to a matrix algebra is "completely positive".Comment: v1: 34 pages, v2: 41 pages; supplementary material is available in the source file, or see http://srag.fmf.uni-lj.si

Matrix Convex Hulls of Free Semialgebraic Sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat C$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, Tarski's transfer principle fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the free (matrix) convex hull of free semialgebraic sets. This article presents the construction of a sequence $C^{(d)}$ of LMI domains in increasingly many variables whose projections $\hat C^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well known classical (commutative) convex hulls does not produce the convex hulls in the free case. This failure is illustrated on one of the simplest free nonconvex $D_p$. A basic question is which free sets $\hat S$ are the projection of a free semialgebraic set $S$? Techniques and results of this paper bear upon this question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a Mathematica notebook) can be found at http://www.math.auckland.ac.nz/~igorklep/publ.htm

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form D_L. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s, f_j and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.Comment: 22 page

Ground state for a massive scalar field in BTZ spacetime with Robin boundary conditions

We consider a real, massive scalar field in BTZ spacetime, a 2+1-dimensional black hole solution of the Einstein's field equations with a negative cosmological constant. First, we analyze the space of classical solutions in a mode decomposition and we characterize the collection of all admissible boundary conditions of Robin type which can be imposed at infinity. Secondly, we investigate whether, for a given boundary condition, there exists a ground state by constructing explicitly its two-point function. We demonstrate that for a subclass of the boundary conditions it is possible to construct a ground state that locally satisfies the Hadamard property. In all other cases, we show that bound state mode solutions exist and, therefore, such construction is not possible.Comment: 17 pages, 3 figure