Beyond Convex Relaxation: A Polynomial–Time Non–Convex Optimization Approach to Network Localization Senshan Ji * , Kam–Fung Sze * , Zirui Zhou * , Anthony Man–Cho So * and Yinyu Ye † * Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong Shatin, N. T., Hong Kong Email: {ssji,kfsze,zrzhou,manchoso}@se.cuhk.edu.hk † Department of Management Science & Engineering Stanford University Stanford, CA 94305 Email: yyye@stanford.edu Abstract—The successful deployment and operation of location–aware networks, which have recently found many ap- plications, depends crucially on the accurate localization of the nodes. Currently, a powerful approach to localization is that of convex relaxation. In a typical application of this approach, the localization problem is first formulated as a rank–constrained semidefinite program (SDP), where the rank corresponds to the target dimension in which the nodes should be localized. Then, the non–convex rank constraint is either dropped or replaced by a convex surrogate, thus resulting in a convex optimization problem. In this paper, we explore the use of a non–convex sur- rogate of the rank function, namely the so–called Schatten quasi– norm, in network localization. Although the resulting optimization problem is non–convex, we show, for the first time, that a first– order critical point can be approximated to arbitrary accuracy in polynomial time by an interior–point algorithm. Moreover, we show that such a first–order point is already sufficient for recovering the node locations in the target dimension if the input instance satisfies certain established uniqueness properties in the literature. Finally, our simulation results show that in many cases, the proposed algorithm can achieve more accurate localization results than standard SDP relaxations of the problem. I. I NTRODUCTION Determining the locations of nodes is a fundamental task in many wireless network applications. From target tracking [1] to emergency response [2], from logistics support [3] to mobile advertising [4], the information collected or transmitted by a node depends crucially on its location. As it is typically impractical to manually position the nodes or equip them with Global Positioning System (GPS) receivers, a key re- search question is how signal metrics (such as received signal strength, time of arrival, angle of arrival, etc. [5]) obtained by individual nodes through direct communication with their neighbors can be used to localize the entire network. One of the most common settings under which the above question is studied is when distances between neighboring nodes can be measured or estimated (this can be achieved using various ranging techniques; see, e.g., [3]). Under this setting, the network localization problem becomes that of determining the node positions in R 2 or R 3 so that they are consistent with the given distance measurements. As is well known, such a fixed– dimensional localization problem is intractable in general [6]. In fact, the localization problem in R 2 remains intractable even when it is known a priori that there is a unique (up to congruences) solution on the plane [7]. Consequently, there has been significant research effort in developing algorithms that can accurately and efficiently localize the nodes in a given dimension; see, e.g., [8] and the references therein. One approach is to use convex relaxation techniques to tackle the intractability of the fixed–dimensional localization problem. This was first adopted by Doherty et al. [9] and has since been extensively developed in the literature; see, e.g., [10]–[19]. In particular, it is now known that such an approach has many desirable features, including polynomial–time solvability and high localization accuracy. However, due to the intractability of the fixed–dimensional localization problem, existing convex relaxation–based localization algorithms will most likely not be able to localize all input instances in the required dimension in polynomial time. Thus, it is natural to ask whether those algorithms can be enhanced so that they can have better localization performance and yet retain their efficiency. To address the above question, it is instructive to start by revisiting the semidefinite programming (SDP) relaxation proposed by Biswas and Ye [20]. The crucial observation underlying Biswas and Ye’s derivation is that the fixed– dimensional localization problem can be formulated as a rank– constrained SDP feasibility problem, i.e., problem of the form find Z such that E (Z )= u, rank(Z )= d, Z symmetric and positive semidefinite. (1) Here, the linear operator E and vector u are determined by the available distance measurements, and d ≥ 1 is the target dimension in which the input instance should be localized (see Section II for details). Thus, by dropping the non–convex rank constraint, one immediately obtains an SDP relaxation of the fixed–dimensional localization problem. As it turns out, the Biswas–Ye SDP relaxation has a nice geometric interpretation. Specifically, So and Ye [11] showed that there is