20,926 research outputs found
An extension of Yuan's Lemma and its applications in optimization
We prove an extension of Yuan's Lemma to more than two matrices, as long as
the set of matrices has rank at most 2. This is used to generalize the main
result of [A. Baccari and A. Trad. On the classical necessary second-order
optimality conditions in the presence of equality and inequality constraints.
SIAM J. Opt., 15(2):394--408, 2005], where the classical necessary second-order
optimality condition is proved under the assumption that the set of Lagrange
multipliers is a bounded line segment. We prove the result under the more
general assumption that the hessian of the Lagrangian evaluated at the vertices
of the Lagrange multiplier set is a matrix set with at most rank 2. We apply
the results to prove the classical second-order optimality condition to
problems with quadratic constraints and without constant rank of the jacobian
matrix
On the (non)existence of best low-rank approximations of generic IxJx2 arrays
Several conjectures and partial proofs have been formulated on the
(non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We
analyze this problem using the Generalized Schur Decomposition and prove
(non)existence of a best rank-R approximation for generic IxJx2 arrays, for all
values of I,J,R. Moreover, for cases where a best rank-R approximation exists
on a set of positive volume only, we provide easy-to-check necessary and
sufficient conditions for the existence of a best rank-R approximation
Three-point bounds for energy minimization
Three-point semidefinite programming bounds are one of the most powerful
known tools for bounding the size of spherical codes. In this paper, we use
them to prove lower bounds for the potential energy of particles interacting
via a pair potential function. We show that our bounds are sharp for seven
points in RP^2. Specifically, we prove that the seven lines connecting opposite
vertices of a cube and of its dual octahedron are universally optimal. (In
other words, among all configurations of seven lines through the origin, this
one minimizes energy for all potential functions that are completely monotonic
functions of squared chordal distance.) This configuration is the only known
universal optimum that is not distance regular, and the last remaining
universal optimum in RP^2. We also give a new derivation of semidefinite
programming bounds and present several surprising conjectures about them.Comment: 30 page
Optimal decomposable witnesses without the spanning property
One of the unsolved problems in the characterization of the optimal
entanglement witnesses is the existence of optimal witnesses acting on
bipartite Hilbert spaces H_{m,n}=C^m\otimes C^n such that the product vectors
obeying =0 do not span H_{m,n}. So far, the only known examples of
such witnesses were found among indecomposable witnesses, one of them being the
witness corresponding to the Choi map. However, it remains an open question
whether decomposable witnesses exist without the property of spanning. Here we
answer this question affirmatively, providing systematic examples of such
witnesses. Then, we generalize some of the recently obtained results on the
characterization of 2\otimes n optimal decomposable witnesses [R. Augusiak et
al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces H_{m,n}
with m,n\geq 3.Comment: 11 pages, published version, title modified, some references added,
other minor improvement
On Marton's inner bound for broadcast channels
Marton's inner bound is the best known achievable region for a general
discrete memoryless broadcast channel. To compute Marton's inner bound one has
to solve an optimization problem over a set of joint distributions on the input
and auxiliary random variables. The optimizers turn out to be structured in
many cases. Finding properties of optimizers not only results in efficient
evaluation of the region, but it may also help one to prove factorization of
Marton's inner bound (and thus its optimality). The first part of this paper
formulates this factorization approach explicitly and states some conjectures
and results along this line. The second part of this paper focuses primarily on
the structure of the optimizers. This section is inspired by a new binary
inequality that recently resulted in a very simple characterization of the
sum-rate of Marton's inner bound for binary input broadcast channels. This
prompted us to investigate whether this inequality can be extended to larger
cardinality input alphabets. We show that several of the results for the binary
input case do carry over for higher cardinality alphabets and we present a
collection of results that help restrict the search space of probability
distributions to evaluate the boundary of Marton's inner bound in the general
case. We also prove a new inequality for the binary skew-symmetric broadcast
channel that yields a very simple characterization of the entire Marton inner
bound for this channel.Comment: Submitted to ISIT 201
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