Escaping Local Optima in a Class of Multi-Agent Distributed Optimization Problems: A Boosting Function Approach

We address the problem of multiple local optima commonly arising in optimization problems for multi-agent systems, where objective functions are nonlinear and nonconvex. For the class of coverage control problems, we propose a systematic approach for escaping a local optimum, rather than randomly perturbing controllable variables away from it. We show that the objective function for these problems can be decomposed to facilitate the evaluation of the local partial derivative of each node in the system and to provide insights into its structure. This structure is exploited by defining "boosting functions" applied to the aforementioned local partial derivative at an equilibrium point where its value is zero so as to transform it in a way that induces nodes to explore poorly covered areas of the mission space until a new equilibrium point is reached. The proposed boosting process ensures that, at its conclusion, the objective function is no worse than its pre-boosting value. However, the global optima cannot be guaranteed. We define three families of boosting functions with different properties and provide simulation results illustrating how this approach improves the solutions obtained for this class of distributed optimization problems

Sensor Deployment for Network-like Environments

This paper considers the problem of optimally deploying omnidirectional sensors, with potentially limited sensing radius, in a network-like environment. This model provides a compact and effective description of complex environments as well as a proper representation of road or river networks. We present a two-step procedure based on a discrete-time gradient ascent algorithm to find a local optimum for this problem. The first step performs a coarse optimization where sensors are allowed to move in the plane, to vary their sensing radius and to make use of a reduced model of the environment called collapsed network. It is made up of a finite discrete set of points, barycenters, produced by collapsing network edges. Sensors can be also clustered to reduce the complexity of this phase. The sensors' positions found in the first step are then projected on the network and used in the second finer optimization, where sensors are constrained to move only on the network. The second step can be performed on-line, in a distributed fashion, by sensors moving in the real environment, and can make use of the full network as well as of the collapsed one. The adoption of a less constrained initial optimization has the merit of reducing the negative impact of the presence of a large number of local optima. The effectiveness of the presented procedure is illustrated by a simulated deployment problem in an airport environment

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although their convergence to the Pareto optimal front (PF) is in several instances superior to that of Pareto-based algorithms, the problem of selecting a way to distribute or guide these solutions in a high-dimensional space has not been explored. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying evolutionary algorithm toward specific regions of interest or the entire Pareto front with the desired distribution of Pareto optimal solutions. Additionally, it is shown that generalized decomposition simplifies many-objective problems by unifying the three performance objectives of multi-objective evolutionary algorithms – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method (CE), is created to illustrate the benefits of the proposed concept for many objective problems. This choice of EDA also enables the test of the hypothesis that low-order statistics based EDAs can have comparable performance to more elaborate EDAs

Visibility-based coverage of mobile sensors in non-convex domains

The area coverage problem of mobile sensor networks has attracted much attention recently, as mobile sensors find many important applications in remote and hostile environments. However, the deployment of mobile sensors in a non-convex domain is nontrivial due to the more general shape of the domain and the attenuation of sensing capabilities caused by the boundary walls or obstacles. We consider the problem of exploration and coverage by mobile sensors in an unknown non-convex domain. We propose the definition of 'visibility-based Voronoi diagram' and extend the continuous-time Lloyd's method, which only works for convex domains, to deploy the mobile sensors in the unknown environments in a distributed manner. Our simulations show the effectiveness of the proposed algorithms. © 2011 IEEE.published_or_final_versionThe 8th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD2011), Qingdao, China, 28-30 June 2011. In Proceedings of the 8th ISVD, 2011, p. 105-11

Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization

Principal component analysis (PCA) is widely used for dimensionality reduction, with well-documented merits in various applications involving high-dimensional data, including computer vision, preference measurement, and bioinformatics. In this context, the fresh look advocated here permeates benefits from variable selection and compressive sampling, to robustify PCA against outliers. A least-trimmed squares estimator of a low-rank bilinear factor analysis model is shown closely related to that obtained from an $\ell_0$-(pseudo)norm-regularized criterion encouraging sparsity in a matrix explicitly modeling the outliers. This connection suggests robust PCA schemes based on convex relaxation, which lead naturally to a family of robust estimators encompassing Huber's optimal M-class as a special case. Outliers are identified by tuning a regularization parameter, which amounts to controlling sparsity of the outlier matrix along the whole robustification path of (group) least-absolute shrinkage and selection operator (Lasso) solutions. Beyond its neat ties to robust statistics, the developed outlier-aware PCA framework is versatile to accommodate novel and scalable algorithms to: i) track the low-rank signal subspace robustly, as new data are acquired in real time; and ii) determine principal components robustly in (possibly) infinite-dimensional feature spaces. Synthetic and real data tests corroborate the effectiveness of the proposed robust PCA schemes, when used to identify aberrant responses in personality assessment surveys, as well as unveil communities in social networks, and intruders from video surveillance data.Comment: 30 pages, submitted to IEEE Transactions on Signal Processin

Connectivity in Dense Networks Confined within Right Prisms

We consider the probability that a dense wireless network confined within a given convex geometry is fully connected. We exploit a recently reported theory to develop a systematic methodology for analytically characterizing the connectivity probability when the network resides within a convex right prism, a polyhedron that accurately models many geometries that can be found in practice. To maximize practicality and applicability, we adopt a general point-to-point link model based on outage probability, and present example analytical and numerical results for a network employing $2 \times 2$ multiple-input multiple-output (MIMO) maximum ratio combining (MRC) link level transmission confined within particular bounding geometries. Furthermore, we provide suggestions for extending the approach detailed herein to more general convex geometries.Comment: 8 pages, 6 figures. arXiv admin note: text overlap with arXiv:1201.401