14 research outputs found
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 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
Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations
In this paper we study feasibility and infeasibility of nonlinear two-stage
fully adjustable robust feasibility problems with an empty first stage. This is
equivalent to deciding whether the uncertainty set is contained within the
projection of the feasible region onto the uncertainty-space. Moreover, the
considered sets are assumed to be described by polynomials. For answering this
question, two very general approaches using methods from polynomial
optimization are presented - one for showing feasibility and one for showing
infeasibility. The developed methods are approximated through sum of squares
polynomials and solved using semidefinite programs. Deciding robust feasibility
and infeasibility is important for gas network operations, which is a nonconvex
feasibility problem where the feasible set is described by a composition of
polynomials with the absolute value function. Concerning the gas network
problem, different topologies are considered. It is shown that a tree
structured network can be decided exactly using linear programming.
Furthermore, a method is presented to reduce a tree network with one additional
arc to a single cycle network. In this case, the problem can be decided by
eliminating the absolute value functions and solving the resulting linearly
many polynomial optimization problems. Lastly, the effectivity of the methods
is tested on a variety of small cyclic networks. It turns out that for
instances where robust feasibility or infeasibility can be decided
successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically
is sufficient