Convex Analysis and Nonlinear Optimization

Borwein, Jonathan M.; Lewis, Adrian S.

doi:10.1007/978-1-4757-9859-3

Cited by 464 publications

(326 citation statements)

References 75 publications

Supporting

Mentioning

320

Contrasting

Unclassified

Order By: Relevance

“…We obtain similarly that (26) holds because δ(σ 2 ) is the solution of the canonical Equation (21). In sum, it appears that (25,26) are equivalent to the canonical Equations 19 and 20. The derivative w.r.t.…”

Section: The Case Q = Isupporting

confidence: 69%

See 1 more Smart Citation

Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

Dupuy

Loubaton²

2012

Entropy

View full text Add to dashboard Cite

This paper provides a comprehensive introduction of large random matrix methods for input covariance matrix optimization of mutual information of MIMO systems. It is first recalled informally how large system approximations of mutual information can be derived. Then, the optimization of the approximations is discussed, and important methodological points that are not necessarily covered by the existing literature are addressed, including the strict concavity of the approximation, the structure of the argument of its maximum, the accuracy of the large system approach with regard to the number of antennas, or the justification of iterative water-filling optimization algorithms. While the existing papers have developed methods adapted to a specific model, this contribution tries to provide a unified view of the large system approximation approach.

show abstract

Section: The Case Q = Isupporting

confidence: 69%

“…and thus coincides with −t δ(σ 2 )δ(σ 2 ) by (25,26). Using (21,22) as well as the expressions (19,20) of…”

Section: The Case Q = Imentioning

confidence: 88%

Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

Dupuy

Loubaton²

2012

Entropy

View full text Add to dashboard Cite

show abstract

“…Furthermore, ss is twice continuously differentiable because the above symmetric function g is twice continuously differentiable at [y]. We can derive the gradient and Hessian of ss following the general formulas for spectral functions, as given in [4,Section 5.2,15]. In this paper, however, we derive simple expressions for the gradient and Hessian by directly applying the chain rule; see Section 3.…”

Section: (G • )(Qw Q T ) = (G • )(W )mentioning

confidence: 99%

“…These methods have well known advantages and disadvantages in speed of convergence, computational cost per iteration, and storage requirements; see, e.g., [16,1,7,21]. These algorithms must be initialized with a point, such as the Metropolis-Hastings weight (4), that satisfies f (w) < ∞. At each step of these algorithms, we need to compute the gradient ∇f (w), and for Newton's method, the Hessian ∇ 2 f (w) as well.…”

Section: Solving the Lmsc Problemmentioning

confidence: 99%

Distributed average consensus with least-mean-square deviation

Xiao

Boyd

Kim

2007

Journal of Parallel and Distributed Computing

976

732

View full text Add to dashboard Cite

We consider a stochastic model for distributed average consensus, which arises in applications such as load balancing for parallel processors, distributed coordination of mobile autonomous agents, and network synchronization. In this model, each node updates its local variable with a weighted average of its neighbors' values, and each new value is corrupted by an additive noise with zero mean. The quality of consensus can be measured by the total mean-square deviation of the individual variables from their average, which converges to a steady-state value. We consider the problem of finding the (symmetric) edge weights that result in the least mean-square deviation in steady state. We show that this problem can be cast as a convex optimization problem, so the global solution can be found efficiently. We describe some computational methods for solving this problem, and compare the weights and the mean-square deviations obtained by this method and several other weight design methods.

show abstract

“…it is invariant to any permutation of the vector entries x i . Hence, it follows from the theory of convex spectral functions that also J is a convex function [2]. Theorem 3.1 states that (15) is a convex problem implying thus that the solution of (15) is unique and that it can be performed efficiently by suitable numeric algorithms.…”

Section: Remarkmentioning

confidence: 99%

Distributed Kalman filtering using consensus strategies

Carli

Chiuso

Schenato

et al. 2007

2007 46th IEEE Conference on Decision and Control

View full text Add to dashboard Cite

Abstract-In this paper, we consider the problem of estimating the state of a dynamical system from distributed noisy measurements. Each agent constructs a local estimate based on its own measurements and estimates from its neighbors. Estimation is performed via a two stage strategy, the first being a Kalman-like measurement update which does not require communication, and the second being an estimate fusion using a consensus matrix. In particular we study the interaction between the consensus matrix, the number of messages exchanged per sampling time, and the Kalman gain. We prove that optimizing the consensus matrix for fastest convergence and using the centralized optimal gain is not necessarily the optimal strategy if the number of message exchange per sampling time is small. Moreover, we prove that under certain conditions the optimal consensus matrix should be doubly stochastic. We also provide some numerical examples to clarify some of the analytical results. I. INTRODUCTIONThe recent technological advances in wireless communication and the decreasing in cost and size of electronic devices, are promoting the appearance of large inexpensive interconnected systems, each with computational and sensing capabilities. These complex systems of agents can be used for monitoring very large scale areas with fine resolution. However, collecting measurements from distributed wireless sensors nodes at a single location for on-line data processing may not be feasible due to several reasons among which long packet delay (e.g. due to multi-hop transmission) and/or limited bandwidth of the wireless network, due e.g. to energy consumption requirements.This problem is apparent in wireless ad-hoc sensor networks where information needs to be multi-hopped from one node to another using closer neighbors. Therefore there is a growing need for in-network data processing tools and algorithms that provide high performance in terms of on-line estimation while (i) reducing the communication load among all sensor nodes, (ii) being very robust to sensor node failures or replacements and packet losses, and (iii) being suitable for distributed control applications.In this work we will focus on distributed estimation of dynamical systems for which sensor nodes are not physically collocated and can communicate with each other according to some underlying communication network. For example, suppose that we want to estimate the temperature in a building that changes according to a random walk, i.e T (t + 1) = T (t) + w(t), where w(t) is a zero-mean random variable with covariance q, and we have N sensors that can measure temperature corrupted by some noise, i.e. y i (t) =

show abstract

Convex Analysis and Nonlinear Optimization

Cited by 464 publications

References 75 publications

Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

Distributed average consensus with least-mean-square deviation

Distributed Kalman filtering using consensus strategies

Contact Info

Product

Resources

About