Generalized Trust Region Subproblem Research Articles

GPS localization problem: a new model and its global optimization

We establish a new fractional squared least squares (FSLS) optimization model for the GPS localization problem. It provides more accurate solutions than the classical squared least squares model. We reformulate (FSLS) as a univariate optimization, where the functional evaluation corresponds to the generalized trust region subproblem. We employ the branch and bound algorithm to globally solve (FSLS) and establish the convergence. It further motivates a much faster iterative heuristic algorithm. Numerical examples are presented to show the accuracy of the new model (FSLS) and the efficiency of the two algorithms.

A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem

<p style='text-indent:20px;'>This paper proposes a second-order cone programming (SOCP) relaxation for the generalized trust-region problem by exploiting the property that any symmetric matrix and identity matrix can be simultaneously diagonalizable. We show that our proposed SOCP relaxation can provide a lower bound as tight as that of the standard semidefinite programming (SDP) relaxation. Moreover, we provide a sufficient condition under which the proposed SOCP relaxation is exact. Since the standard SDP relaxation suffers from a much heavier computing burden, the proposed SOCP relaxation has a much higher efficiency in solving process. Then we design a branch-and-bound algorithm based on this SOCP relaxation to obtain the global optimal solution for a general problem. Three types of numerical experiments are carried out to demonstrate the effectiveness and efficiency of our proposed SOCP relaxation.

Noisy Euclidean distance matrix completion with a single missing node

We present several solution techniques for the noisy single source localization problem, i.e. the Euclidean distance matrix completion problem with a single missing node to locate under noisy data. For the case that the sensor locations are fixed, we show that this problem is implicitly convex, and we provide a purification algorithm along with the SDP relaxation to solve it efficiently and accurately. For the case that the sensor locations are relaxed, we study a model based on facial reduction. We present several approaches to solve this problem efficiently, and we compare their performance with existing techniques in the literature. Our tools are semidefinite programming, Euclidean distance matrices, facial reduction, and the generalized trust region subproblem. We include extensive numerical tests.

A Robust, Non-Cooperative Localization Algorithm in the Presence of Outlier Measurements in Ocean Sensor Networks.

As an important means of multidimensional observation on the sea, ocean sensor networks (OSNs) could meet the needs of comprehensive information observations in large-scale and multifactor marine environments. In what concerns OSNs, accurate location information is the basis of the data sets. However, because of the multipath effect—signal shadowing by waves and unintentional or malicious attacks—outlier measurements occur frequently and inevitably, which directly degrades the localization accuracy. Therefore, increasing localization accuracy in the presence of outlier measurements is a critical issue that needs to be urgently tackled in OSNs. In this case, this paper proposed a robust, non-cooperative localization algorithm (RNLA) using received signal strength indication (RSSI) in the presence of outlier measurements in OSNs. We firstly formulated the localization problem using a log-normal shadowing model integrated with a first order Taylor series. Nevertheless, the problem was infeasible to solve, especially in the presence of outlier measurements. Hence, we then converted the localization problem into the optimization problem using squared range and weighted least square (WLS), albeit in a nonconvex form. For the sake of an accurate solution, the problem was then transformed into a generalized trust region subproblem (GTRS) combined with robust functions. Although GTRS was still a nonconvex framework, the solution could be acquired by a bisection approach. To ensure global convergence, a block prox-linear (BPL) method was incorporated with the bisection approach. In addition, we conducted the Cramer–Rao low bound (CRLB) to evaluate RNLA. Simulations were carried out over variable parameters. Numerical results showed that RNLA outperformed the other algorithms under outlier measurements, notwithstanding that the time for RNLA computation was a little bit more than others in some conditions.

Solution and Analysis of TDOA Localization of a Near or Distant Source in Closed Form

Point positioning and direction of arrival (DOA) localization require separate estimators, depending on whether the source is near or distant from the sensors. The use of modified polar representation (MPR) for the source location enables the integration of the two estimators together and eliminates the knowledge needed if the source is in the near-field or far-field. The previous work on MPR only provides an iterative implementation of the maximum likelihood estimator (MLE) initialized by a coarse semidefinite relaxation (SDR) solution. This paper proposes a formulation for time difference of arrival (TDOA) localization of a 2-D or 3-D source in MPR that would lead to a closed-form solution through the minimization of a quadratic function with a quadratic constraint. Two techniques, the successive unconstrained minimization (SUM) and the generalized trust region subproblem (GTRS), are applied to solve the optimization. Detailed analysis in the first order for mean-square and in the second order for estimation bias is performed for both methods. The theoretical results illustrate that the closed-form solution from each of the methods provides the Cramer–Rao lower bound performance for Gaussian noise. The GTRS solution has better accuracy than the SUM solution when the source signal is arriving at an azimuth or elevation angle close to zero, 90 $^\circ$ , or 180 $^\circ$ or when the measurement noise level is large. They yield comparable performance with the MLE albeit having less complexity and avoiding possible divergence behavior. They outperform the closed-form solutions from the literature in angle estimation and have smaller bias.

A Robust NLOS Bias Mitigation Technique for RSS-TOA-Based Target Localization

This letter proposes a novel robust mitigation technique to address the problem of target localization in adverse non-line-of-sight (NLOS) environments. The proposed scheme is based on combined received signal strength and time of arrival measurements. Influence of NLOS biases is mitigated by treating them as nuisance parameters through a robust approach. Due to a high degree of difficulty of the considered problem, it is converted into a generalized trust region sub-problem by applying certain approximations, and solved efficiently by merely a bisection procedure. Numerical results corroborate the effectiveness of the proposed approach, rendering it the most accurate one in all considered scenarios.

Multistatic Sonar Localization With a Transmitter

This paper focuses on the problem of multistatic sonar localization with a transmitter where the known signal transmission speed, the transmitter and receiver positions are all contaminated by Gaussian noise. The Cramer-Rao lower bound (CRLB) are derived for the object position and the localization parameters. The analysis in CRLB shows that the localization parameters have a chance to be updated to more accurate ones. Two solutions are then proposed to estimate the object position using the time measurements and angle measurements. One is a two-step closed-form solution based on weighting least squares and the other is a generalized trust region subproblem (GTRS) solution using Newton’s method. A recursive MLE is also proposed to update the localization parameters and a more accurate propagation speed can be obtained from the proposed MLE especially when the propagation speed noise is large. Simulations show that the two localization solutions and the propagation speed updated from the proposed MLE can reach their CRLBs.

Novel Reformulations and Efficient Algorithms for the Generalized Trust Region Subproblem

We present a new solution framework to solve the generalized trust region subproblem (GTRS) of minimizing a quadratic objective over a quadratic constraint. More specifically, we derive a convex qu...

Exact Robust Solution to TW-ToA-Based Target Localization Problem With Clock Imperfections

This letter addresses the problem of target localization based on two-way time of arrival (TW-ToA) measurements with clock imperfections. In addition to the target location, the turn-around times and clock skews are considered unknown. Since an optimal estimator for this problem cannot be tackled directly, we approximate it by a suboptimal, robust one, formulated as a generalized trust region subproblem. Even though nonconvex in general, exact solution of the derived estimator can be obtained by just a bisection procedure. Simulation results validate the effectiveness of the proposed technique, matching the performance of the state of the art with significantly lower computational complexity.

Target Localization in Underwater Acoustic Sensor Networks Using RSS Measurements

This paper addresses the target localization problems based on received signal strength (RSS) measurements in underwater acoustic wireless sensor network (UWSN). Firstly, the problems based on the maximum likelihood (ML) criterion for estimating target localization in cases of both known and unknown transmit power are respectively derived, and fast implementation algorithms are proposed by transforming the non-convex problems into a generalized trust region subproblem (GTRS) frameworks. A three-step procedure is also provided to enhance the estimation accuracy in the unknown target transmit power case. Furthermore, the Cramer–Rao lower bounds (CRLBs) in both cases are derived. Computer simulation results show the superior performance of the proposed methods in the underwater environment.

A bisection-based approach for exact target localization in NLOS environments

This work addresses the range-based target localization problem in adverse non-line-of-sight (NLOS) environments. We start by deriving the maximum likelihood (ML) estimator from the measurement model, since it is asymptotically efficient. However, this estimator is highly non-convex and difficult to solve directly. Hence, we convert the localization problem into a generalized trust region sub-problem (GTRS) framework. Although still non-convex in general, the derived estimator is strictly decreasing over a readily obtained interval, and thus, can be solved exactly by a bisection procedure. In huge contrast to existing algorithms, which either require the knowledge about the magnitude of the NLOS bias or to a priori distinguish between line-of-sight (LOS) and NLOS links, the new one does not require such prerequisites. Also, the computational complexity of the proposed algorithm is linear in the number of reference nodes, unlike the majority of existing ones. Our simulation results show that the new algorithm possesses a steady NLOS bias mitigation capacity and that it represents an excellent alternative in the sense of the trade off between accuracy and complexity. To be more specific, it not only matches the performance of existing methods (majority of which significantly more computationally complex) but outperforms them in general. Moreover, the performance of the proposed algorithm is validated through real-indoor experimental data.

SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices

We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and Hertog (Math. Program. 143(1-2):1-29, 2014), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. Compared with the state-of-the-art approach to reformulate the GTRS as a semi-definite programming problem, our SOCP reformulation delivers a much faster solution algorithm. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint. Our methods also enable us to obtain simplified versions of the classical S-lemma, the S-lemma with equality, and the S-lemma with interval bounds.

3-D Target Localization in Wireless Sensor Networks Using RSS and AoA Measurements

This paper addresses target localization problems in both noncooperative and cooperative 3-D wireless sensor networks (WSNs), for both cases of known and unknown sensor transmit power, i.e., $P_{T}$ . We employ a hybrid system that fuses distance and angle measurements, extracted from the received signal strength and angle-of-arrival information, respectively. Based on range and angle measurement models, we derive a novel nonconvex estimator based on the least squares criterion. The derived nonconvex estimator tightly approximates the maximum-likelihood estimator for small noise. We then show that the developed estimator can be transformed into a generalized trust region subproblem framework, by following the squared range approach, for noncooperative WSNs. For cooperative WSNs, we show that the estimator can be transformed into a convex problem by applying appropriate semidefinite programming relaxation techniques. Moreover, we show that the generalization of the proposed estimators for known $P_{T}$ is straightforward to the case where $P_{T}$ is not known. Our simulation results show that the new estimators have excellent performance and are robust to not knowing $P_{T}$ . The new estimators for noncooperative localization significantly outperform the existing estimators, and our estimators for cooperative localization show exceptional performance in all considered settings.

Distributed algorithm for target localization in wireless sensor networks using RSS and AoA measurements

This paper addresses target localization problem in a cooperative 3-D wireless sensor network (WSN). We employ a hybrid system that fuses distance and angle measurements, extracted from the received signal strength (RSS) and angle-of-arrival (AoA) information, respectively. Based on range measurement model and simple geometry, we derive a novel non-convex estimator based on the least squares (LS) criterion. The derived non-convex estimator tightly approximates the maximum likelihood (ML) one for small noise levels. We show that the developed non-convex estimator is suitable for distributed implementation, and that it can be transformed into a convex one by applying a second-order cone programming (SOCP) relaxation technique. We also show that the developed non-convex estimator can be transformed into a generalized trust region sub-problem (GTRS) framework, by following the squared range (SR) approach. The proposed SOCP algorithm for known transmit powers is then generalized to the case where the transmit powers are different and not known. Furthermore, we provide a detailed analysis of the computational complexity of the proposed algorithms. Our simulation results show that the new estimators have excellent performance in terms of the estimation accuracy and convergence, and they confirm the effectiveness of combining two radio measurements.

An efficient algorithm for solving the generalized trust region subproblem

In this paper, we consider the interval bounded generalized trust region subproblem (GTRS) which is the problem of minimizing a general quadratic function subject to an upper and lower bounded general quadratic constraint. Under the assumption that two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence, a diagonalization-based algorithm is introduced to solve it by showing that GTRS is indeed equivalent to a linearly constrained convex univariate problem. Some numerical experiments are given to show the effectiveness of the proposed method and to compare it with the extended Rendl–Wolkowicz algorithm due to Pong and Wolkowicz.

Spatial Sound Localization via Multipath Euclidean Distance Matrix Recovery

A novel localization approach is proposed in order to find the position of an individual source using recordings of a single microphone in a reverberant enclosure. The multipath propagation is modeled by multiple virtual microphones as images of the actual single microphone and a multipath distance matrix is constructed whose components consist of the squared distances between the pairs of microphones (real or virtual) or the squared distances between the microphones and the source. The distances between the actual and virtual microphones are computed from the geometry of the enclosure. The microphone-source distances correspond to the support of the early reflections in the room impulse response associated with the source signal acquisition. The low-rank property of the Euclidean distance matrix is exploited to identify this correspondence. Source localization is achieved through optimizing the location of the source matching those measurements. The recording time of the microphone and generation of the source signal is asynchronous and estimated via the proposed procedure. Furthermore, a theoretically optimal joint localization and synchronization algorithm is derived by formulating the source localization as minimization of a quartic cost function. It is shown that the global minimum of the proposed cost function can be efficiently computed by converting it to a generalized trust region sub-problem. Numerical simulations on synthetic data and real data recordings obtained by practical tests show the effectiveness of the proposed approach.

Strong duality for generalized trust region subproblem: S-lemma with interval bounds

With the help of the newly developed S-lemma with interval bounds, we show that strong duality holds for the interval bounded generalized trust region subproblem under some mild assumptions, which answers an open problem raised by Pong and Wolkowicz [Comput. Optim. Appl. 58(2), 273-322, 2014].

The generalized trust region subproblem

The interval bounded generalized trust region subproblem (GTRS) consists in minimizing a general quadratic objective, q 0(x)?min, subject to an upper and lower bounded general quadratic constraint, l≤q 1(x)≤u. This means that there are no definiteness assumptions on either quadratic function. We first study characterizations of optimality for this implicitly convex problem under a constraint qualification and show that it can be assumed without loss of generality. We next classify the GTRS into easy case and hard case instances, and demonstrate that the upper and lower bounded general problem can be reduced to an equivalent equality constrained problem after identifying suitable generalized eigenvalues and possibly solving a sparse system. We then discuss how the Rendl-Wolkowicz algorithm proposed in Fortin and Wolkowicz (Optim. Methods Softw. 19(1):41---67, 2004) and Rendl and Wolkowicz (Math. Program. 77(2, Ser. B):273---299, 1997) can be extended to solve the resulting equality constrained problem, highlighting the connection between the GTRS and the problem of finding minimum generalized eigenvalues of a parameterized matrix pencil. Finally, we present numerical results to illustrate this algorithm at the end of the paper.

Quadratic Equality Constrained Tracking Algorithm Using TDOA measurements

Moving target tracking using time difference of arrival measurements has received significant attention in recent years. When the target trajectory satisfies quadratic equality constraints, second-order nonlinear equations can be incorporated in the tracking algorithm to improve the accuracy. Existing methods dealing with constraints may suffer from a lack of convergence or large computation, and are often affected by the initial value of the iteration. The proposed algorithm first utilizes standard Kalman filter to update state estimation and then refines it with a maximum likelihood estimator. When solving the constrained maximum likelihood problem, a kind of generalized trust region sub-problem is incorporated to obtain the global optimal solution. Computer simulation results show that the proposed algorithm outperforms the existing methods in tracking accuracy and do not diverge when the initial state is unknown.

A Lagrangian Dual Approach to the Single-Source Localization Problem

The single-source localization problem (SSLP), which is nonconvex by its nature, appears in several important multidisciplinary fields such as signal processing and the global positioning system. In this paper, we cast SSLP as a Euclidean distance embedding problem and study a Lagrangian dual approach. It is proved that the Lagrangian dual problem must have an optimal solution under the generalized Slater condition. We provide a sufficient condition for the zero-duality gap and establish the equivalence between the Lagrangian dual approach and the existing Generalized Trust-Region Subproblem (GTRS) approach studied by Beck et al. [“Exact and Approximate Solutions of Source Localization Problems,” IEEE Trans. Signal Process., vol. 56, pp. 1770-1778, 2008]. We also reveal new implications of the assumptions made by the GTRS approach. Moreover, the Lagrangian dual approach has a straightforward extension to the multiple-source localization problem. Numerical simulations demonstrate that the Lagrangian dual approach can produce localization of similar quality as the GTRS and can significantly outperform the well-known semidefinite programming solver for the multiple source localization problem on the tested cases.