The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra

This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets { X : L(X) is positive semidefinite }, called free spectrahedra. The Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros-Winkler Theorem. The projection of a free spectrahedron in g+h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.Comment: v2: 56 pages, reworked abstract and intro to emphasize the convex duality aspects; v1: 60 pages; includes an index and table of content

Exposed faces of semidefinitely representable sets

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie

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

Orbitopes

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, in particular convex geometry, optimization, and algebraic geometry. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n). These include Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese orbitopes and Grassmann orbitopes. We study their face lattices, their algebraic boundary hypersurfaces, and representations as spectrahedra or projected spectrahedra.Comment: 37 pages. minor revisions of origina

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