Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Note on Cores and Quasi Relative Interiors in Partially Finite Convex Programming
45
0
0.0
(
0
)
Authors
Scott B. Lindstrom
Ask ChatGPT about the research
No Arabic abstract
The problem of minimizing an entropy functional subject to linear constraints is a useful example of partially finite convex programming. In the 1990s, Borwein and Lewis provided broad and easy-to-verify conditions that guarantee strong duality for such problems. Their approach is to construct a function in the quasi-relative interior of the relevant infinite-dimensional set, which assures the existence of a point in the core of the relevant finite-dimensional set. We revisit this problem, and provide an alternative proof by directly appealing to the definition of the core, rather than by relying on any properties of the quasi-relative interior. Our approach admits a minor relaxation of the linear independence requirements in Borwein and Lewis framework, which allows us to work with certain piecewise-defined moment functions precluded by their conditions. We provide such a computed example that illustrates how this relaxation may be used to tame observed Gibbs phenomenon when the underlying data is discontinuous. The relaxation illustrates the understanding we may gain by tackling partially-finite problems from both the finite-dimensional and infinite-dimensional sides. The comparison of these two approaches is informative, as both proofs are constructive.
rate research
Read More
Let $Rx$ denote the ring of polynomials in $g$ freely non-commuting variables $x=(x_1,...,x_g)$. There is a natural involution * on $Rx$ determined by $x_j^*=x_j$ and $(pq)^*=q^* p^*$ and a free polynomial $pinRx$ is symmetric if it is invariant under this involution. If $X=(X_1,...,X_g)$ is a $g$ tuple of symmetric $ntimes n$ matrices, then the evaluation $p(X)$ is naturally defined and further $p^*(X)=p(X)^*$. In particular, if $p$ is symmetric, then $p(X)^*=p(X)$. The main result of this article says if $p$ is symmetric, $p(0)=0$ and for each $n$ and each symmetric positive definite $ntimes n$ matrix $A$ the set ${X:A-p(X)succ 0}$ is convex, then $p$ has degree at most two and is itself convex, or $-p$ is a hermitian sum of squares.
We analyze a class of sublinear functionals which characterize the interior and the exterior of a convex cone in a normed linear space.
We present a convex-concave reformulation of the reversible Markov chain estimation problem and outline an efficient numerical scheme for the solution of the resulting problem based on a primal-dual interior point method for monotone variational inequalities. Extensions to situations in which information about the stationary vector is available can also be solved via the convex- concave reformulation. The method can be generalized and applied to the discrete transition matrix reweighting analysis method to perform inference from independent chains with specified couplings between the stationary probabilities. The proposed approach offers a significant speed-up compared to a fixed-point iteration for a number of relevant applications.
We study a class of non-unitary so(2,d) representations (for even values of d), describing mixed-symmetry partially massless fields which constitute natural candidates for defining higher-spin singletons of higher order. It is shown that this class of so(2,d) modules obeys of natural generalisation of a couple of defining properties of unitary higher-spin singletons. In particular, we find out that upon restriction to the subalgebra so(2,d-1), these representations branch onto a sum of modules describing partially massless fields of various depths. Finally, their tensor product is worked out in the particular case of d=4, where the appearance of a variety of mixed-symmetry partially massless fields in this decomposition is observed.
A definition of quasi-local energy in a gravitational field based upon its embedding into flat space is discussed. The outcome is not satisfactory from many points of view.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
