Abstract—The matrix rank minimization problem consists of finding a matrix of minimum rank that satisfies given convex constraints. It is NP-hard in general and has applications in control, system identification, and machine learning. Reweighted trace minimization has been considered as an iterative heuristic for this problem. In this paper, we analyze the convergence of this iterative heuristic, showing that the difference between successive iterates tends to zero. Then, after reformulating the heuristic as reweighted nuclear norm minimization, we propose an efficient gradient-based implementation that takes advantage of the new formulation and opens the way to solving largescale problems. We apply this algorithm to the problem of loworder system identification from input-output data. Numerical examples demonstrate that the reweighted nuclear norm minimization makes model order selection easier and results in lower order models compared to nuclear norm minimization without weights. A. Background I.

We study the problem of traffic matrix estimation in large IP networks. The problem of inferring the traffic for each (origin- destination) pair in the network from the link measurements has been widely studied over the past 10 years. Recently, the possibility of using network-monitoring tools such as Netflow (cisco Systems) has renewed the interest for this problem, the aim being now to make the best possible use of such tools. Because of its expensive cost of use and deployment, optimizing the set of interfaces on which Netflow should be installed (and/or activated) is of first interest. We model the optimal Netflow deployment problem in terms of experimental design, which allows us to use convex programming and semidefinite programming techniques. The principle is to choose a set of localizations for Netflow that minimizes (in a certain sense) the variance of an estimator for the traffic matrix. We show that this problem is a generalization of the rank-maximization of the observation matrix, and that the greedy algorithm provides a (polynomial time) approximation of the solution by (1 − 1/e). We give experimental results on real networks and we show that an approach based on the rounding of a continuous relaxation improves on other approaches, including greedy algorithms.

Recent advances in sensor technology coupled with embedded systems and wireless networking has made it possible to deploy sensors for numerous applications including target tracking, environmental science, defense information, and security. Sensor scheduling, a process to allocate sensing resources by optimizing a performance metric over a future time-horizon under constraints, is an effective method to improve performance for such problems. This work investigates myopic (one step ahead) and non-myopic (multiple steps ahead) sensor scheduling algorithms for target tracking applications. Two methods of predicting tracker performance are developed that can be used for tar-get tracking applications. The first is covariance-based, and it can be used with covariance-based scheduler costs. The second is unscented transform-based and it can be used with arbitrary scheduler costs. In application, both methods give a significant improvement in tracking performance over the tracking performance without sensor scheduling. The use of non-myopic sensor scheduling is often restricted due to an exponential dependency of computational and memory requirements on the length of prediction horizon. For

Abstract We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order n ≥ 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order n ≤ 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363–385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.

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