40,095 research outputs found
Inexact Convex Relaxations for AC Optimal Power Flow: Towards AC Feasibility
Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted
significant interest as in several instances they provably yield the global
optimum to the original non-convex problem. If, however, the relaxation is
inexact, the obtained solution is not AC-feasible. The quality of the obtained
solution is essential for several practical applications of AC-OPF, but
detailed analyses are lacking in existing literature. This paper aims to cover
this gap. We provide an in-depth investigation of the solution characteristics
when convex relaxations are inexact, we assess the most promising AC
feasibility recovery methods for large-scale systems, and we propose two new
metrics that lead to a better understanding of the quality of the identified
solutions. We perform a comprehensive assessment on 96 different test cases,
ranging from 14 to 3120 buses, and we show the following: (i) Despite an
optimality gap of less than 1%, several test cases still exhibit substantial
distances to both AC feasibility and local optimality and the newly proposed
metrics characterize these deviations. (ii) Penalization methods fail to
recover an AC-feasible solution in 15 out of 45 cases, and using the proposed
metrics, we show that most failed test instances exhibit substantial distances
to both AC-feasibility and local optimality. For failed test instances with
small distances, we show how our proposed metrics inform a fine-tuning of
penalty weights to obtain AC-feasible solutions. (iii) The computational
benefits of warm-starting non-convex solvers have significant variation, but a
computational speedup exists in over 75% of the cases
On the Analysis of Trajectories of Gradient Descent in the Optimization of Deep Neural Networks
Theoretical analysis of the error landscape of deep neural networks has
garnered significant interest in recent years. In this work, we theoretically
study the importance of noise in the trajectories of gradient descent towards
optimal solutions in multi-layer neural networks. We show that adding noise (in
different ways) to a neural network while training increases the rank of the
product of weight matrices of a multi-layer linear neural network. We thus
study how adding noise can assist reaching a global optimum when the product
matrix is full-rank (under certain conditions). We establish theoretical
foundations between the noise induced into the neural network - either to the
gradient, to the architecture, or to the input/output to a neural network - and
the rank of product of weight matrices. We corroborate our theoretical findings
with empirical results.Comment: 4 pages + 1 figure (main, excluding references), 5 pages + 4 figures
(appendix
Stable Wireless Network Control Under Service Constraints
We consider the design of wireless queueing network control policies with
particular focus on combining stability with additional application-dependent
requirements. Thereby, we consequently pursue a cost function based approach
that provides the flexibility to incorporate constraints and requirements of
particular services or applications. As typical examples of such requirements,
we consider the reduction of buffer underflows in case of streaming traffic,
and energy efficiency in networks of battery powered nodes. Compared to the
classical throughput optimal control problem, such requirements significantly
complicate the control problem. We provide easily verifyable theoretical
conditions for stability, and, additionally, compare various candidate cost
functions applied to wireless networks with streaming media traffic. Moreover,
we demonstrate how the framework can be applied to the problem of energy
efficient routing, and we demonstrate the aplication of our framework in
cross-layer control problems for wireless multihop networks, using an advanced
power control scheme for interference mitigation, based on successive convex
approximation. In all scenarios, the performance of our control framework is
evaluated using extensive numerical simulations.Comment: Accepted for publication in IEEE Transactions on Control of Network
Systems. arXiv admin note: text overlap with arXiv:1208.297
On the Optimality of Treating Inter-Cell Interference as Noise in Uplink Cellular Networks
In this paper, we explore the information-theoretic optimality of treating
interference as noise (TIN) in cellular networks. We focus on uplink scenarios
modeled by the Gaussian interfering multiple access channel (IMAC), comprising
mutually interfering multiple access channels (MACs), each formed by an
arbitrary number of transmitters communicating independent messages to one
receiver. We define TIN for this setting as a scheme in which each MAC (or
cell) performs a power-controlled version of its capacity-achieving strategy,
with Gaussian codebooks and successive decoding, while treating interference
from all other MACs (i.e. inter-cell interference) as noise. We characterize
the generalized degrees-of-freedom (GDoF) region achieved through the proposed
TIN scheme, and then identify conditions under which this achievable region is
convex without the need for time-sharing. We then tighten these convexity
conditions and identify a regime in which the proposed TIN scheme achieves the
entire GDoF region of the IMAC and is within a constant gap of the entire
capacity region.Comment: Accepted for publication in IEEE Transactions on Information Theor
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