1,003 research outputs found
Hidden Convexity in Partially Separable Optimization
The paper identifies classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some partial separable structure. Among other things, the new hidden convexity results open up the possibility to solve multi-stage robust optimization problems using certain nonlinear decision rules.convex relaxation of nonconvex problems;hidden convexity;partially separable functions;robust optimization
Stability in Designer Gravity
We study the stability of designer gravity theories, in which one considers
gravity coupled to a tachyonic scalar with anti-de Sitter boundary conditions
defined by a smooth function W. We construct Hamiltonian generators of the
asymptotic symmetries using the covariant phase space method of Wald et al.and
find they differ from the spinor charges except when W=0. The positivity of the
spinor charge is used to establish a lower bound on the conserved energy of any
solution that satisfies boundary conditions for which has a global minimum.
A large class of designer gravity theories therefore have a stable ground
state, which the AdS/CFT correspondence indicates should be the lowest energy
soliton. We make progress towards proving this, by showing that minimum energy
solutions are static. The generalization of our results to designer gravity
theories in higher dimensions involving several tachyonic scalars is discussed.Comment: 29 page
Towards a Big Crunch Dual
We show there exist smooth asymptotically anti-de Sitter initial data which
evolve to a big crunch singularity in a low energy supergravity limit of string
theory. This opens up the possibility of using the dual conformal field theory
to obtain a fully quantum description of the cosmological singularity. A
preliminary study of this dual theory suggests that the big crunch is an
endpoint of evolution even in the full string theory. We also show that any
theory with scalar solitons must have negative energy solutions. The results
presented here clarify our earlier work on cosmic censorship violation in N=8
supergravity.Comment: 27 pages, 3 figures;v2:minor correction
Holographic Description of AdS Cosmologies
To gain insight in the quantum nature of the big bang, we study the dual
field theory description of asymptotically anti-de Sitter solutions of
supergravity that have cosmological singularities. The dual theories do not
appear to have a stable ground state. One regularization of the theory causes
the cosmological singularities in the bulk to turn into giant black holes with
scalar hair. We interpret these hairy black holes in the dual field theory and
use them to compute a finite temperature effective potential. In our study of
the field theory evolution, we find no evidence for a "bounce" from a big
crunch to a big bang. Instead, it appears that the big bang is a rare
fluctuation from a generic equilibrium quantum gravity state.Comment: 34 pages, 8 figures, v2: minor changes, references adde
Generic Cosmic Censorship Violation in anti de Sitter Space
We consider (four dimensional) gravity coupled to a scalar field with
potential V(\phi). The potential satisfies the positive energy theorem for
solutions that asymptotically tend to a negative local minimum. We show that
for a large class of such potentials, there is an open set of smooth initial
data that evolve to naked singularities. Hence cosmic censorship does not hold
for certain reasonable matter theories in asymptotically anti de Sitter
spacetimes. The asymptotically flat case is more subtle. We suspect that
potentials with a local Minkowski minimum may similarly lead to violations of
cosmic censorship in asymptotically flat spacetimes, but we do not have
definite results.Comment: 4 pages, v2: minor change
Gravitational Wave Bursts from Cosmic Superstrings with Y-junctions
Cosmic superstring loops generically contain strings of different tensions
that meet at Y-junctions. These loops evolve non-periodically in time, and have
cusps and kinks that interact with the junctions. We study the effect of
junctions on the gravitational wave signal emanating from cosmic string cusps
and kinks. We find that earlier results on the strength of individual bursts
from cusps and kinks on strings without junctions remain largely unchanged, but
junctions give rise to additional contributions to the gravitational wave
signal coming from strings expanding at the speed of light at a junction and
kinks passing through a junction.Comment: 20 pages, 5 figure
Hidden Convexity in Partially Separable Optimization
The paper identifies classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some partial separable structure. Among other things, the new hidden convexity results open up the possibility to solve multi-stage robust optimization problems using certain nonlinear decision rules.
- âŠ