A nonsmooth optimization approach to sensor network localization # Adil Bagirov 1 , Daniel T.H. Lai 2 , M. Palaniswami 2 1 School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat Victoria 3353 Australia, email: a.bagirov@ballarat.edu.au 2 Department of Electrical and Electronic Engineering, University of Melbourne, Parkville Campus, Victoria 3010 Australia, email: d.lai, swami@ee.unimelb.edu.au Abstract In this paper the problem of localization of wireless sensor network is formulated as an unconstrained nonsmooth opti- mization problem. We minimize a distance objective function which incorporates unknown sensor nodes and nodes with known positions (anchors) in contrast to popular semidefinite programming (SDP) methods which use artificial objective functions. We study the main properties of the objective function in this problem and design an algorithm for its mini- mization. Our algorithm is a derivative-free discrete gradient method that allows one to find a near global solution. The al- gorithm can handle a large number of sensors in the network. This paper contains the theory of our proposed formulation and algorithm while experimental results are included in later work. Key words and phrases: sensor networks, nonsmooth optimiza- tion, derivative free algorithms. 1. I NTRODUCTION Wireless sensor networks (WSN) have drawn great interest recently, with applications ranging from environmental moni- toring, patient observation in healthcare monitoring to military tracking on battlefields [1], [8]. Sensor networks potentially consist of a large number of sensor nodes (hundreds to thousands) which may be for example, strategically placed in hospitals for patient monitoring or randomly positioned (scattered) e.g., airdropped on a battlefield. One of the major deployment issues of wireless sensor networks is node place- ment which comprises two distinct but related problems. The first problem is known as optimal placement where one has to strategically position a number of nodes to fulfil certain criteria e.g., maximum coverage [22]. The second problem which is the focus of this paper is self-localization, where the node positions are unknown and the network tries to discover their positions [18]. This is achieved using only sensor informa- tion such as received signal strength (RSS), time difference of arrival (TDOA) [17] and hop connectivities [16]. Self- localization is important because sensor data frequently require node positions to be of practical use, for example tracking troop movements on the battlefield requires accurate knowl- edge of sensor positions. While global positioning systems (GPS) are attractive solutions for WSN localization, they are not yet cost effective enough to be deployed on every sensor node and may not function well in enclosed areas with no satellite line of sight. Self-localization is not an easy problem as evidenced by the many existing techniques such as semidefinite programming (SDP) [6], [7], multidimensional scaling (MDS) [10], [19], [20], particle filter modelling [13] and kernel methods [15]. In SDP methods, one assumes that the network consists of N x nodes with unknown positions and N a nodes with known positions called anchors. These methods are easy to implement due to the availability of existing algorithms but suffer from dimensionality problems. The number of SDP variables increases quadratically regardless of the number of unknown nodes and anchor nodes causing larger problems to be difficult to handle. Modifications to the SDP method include relaxation [6] techniques which provide a faster localization algorithm at the cost of an approximate solution. MDS meth- ods are attractive because they do not require anchor nodes, however the algorithms suffer from local minima meaning that algorithm initial conditions determine the final topology estimates. Local minima nevertheless could be handled by ex- pressing node proximities as convex constraints and employing convex programming algorithms [12]. In this paper, we propose another method to handle local minima using a nonsmooth optimization approach. The orig- inal SDP constraints is first transformed to a least squares minimization problem and a derivative free nonsmooth opti- mization algorithm is applied to solve it. Our algorithm can handle nonsmooth objective functions and searches for a better solution as opposed to previous methods which terminate at approximate solutions. The algorithm finds a sequence of local solutions and specifically exploits the problem structure making it applicable for large scale sensor network localization problems. The paper is organized as follows. In Section 2 the non- smooth optimization formulation of wireless sensor network problem is given. The properties of the objective function in 1-4244-1502-0/07/$25.00  2007 IEEE ISSNIP 2007 727