Generalized Trust Region Subproblem Research Articles

Positive semidefinite interval of matrix pencil and its applications to the generalized trust region subproblems

We are concerned with finding the set I⪰(A,B) of real values μ such that the matrix pencil A+μB is positive semidefinite. If A,B are not simultaneously diagonalizable via congruence (SDC), I⪰(A,B) either is empty or has only one value μ. When A,B are SDC, I⪰(A,B), if not empty, can be a singleton or an interval. Especially, if I⪰(A,B) is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval I≻(A,B). If A,B are both singular, then even I⪰(A,B) is an interval, its interior may not be I≻(A,B), but A,B are then decomposed as block diagonals of submatrices A1,B1 with B1 nonsingular such that I⪰(A,B)=I⪰(A1,B1). Applying I⪰(A,B), the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.

Novel High-Precision and High-Robustness Localization Algorithm for Underwater-Environment-Monitoring Wireless Sensor Networks

In marine ecological environment monitoring, the acquisition of node location information is crucial, and the absence of location information can render the collected data meaningless. Compared to the rest of the distance-based localization methods, the received signal strength (RSS)-based localization technique has gained significant interest due to its low cost and the absence of time synchronization. However, the acoustic signal propagates in the complex and changeable aqueous medium, and, in addition to the time-varying path loss factor (PLF), there is often a certain absorption loss, which seriously deteriorates the localization accuracy of the RSS-based technique. To address the above challenges, we propose a novel high-precision and high-robustness localization (NHHL) algorithm that introduces an estimation parameter to conjointly estimate the marine node location and the ambient PLF. Firstly, the original non-convex localization problem is converted into an alternating nonnegative constrained least squares (ANCLS) framework with the unknown PLF and absorption loss, and a two-step localization method based on the primitive dual interior point method and block co-ordinate update method is presented to find the optimal solution. In the first step, the penalty function is utilized to reformulate the localization problem and find an approximate solution. Nevertheless, due to inherent errors, it is unable to approximate the constraint boundary and the global optimum solution. Subsequently, in the second step, the original localization problem is further transformed into a generalized trust region sub-problem (GTRS) framework, and the approximate solution of the interior point method is utilized as the initial estimation, and then iteratively solved by block co-ordinate update to obtain the precise location and PLF conjointly. Furthermore, the closed-form expression of the Cramér–Rao lower bound (CRLB) for the case of the unknown path loss factor and absorption loss is derived to evaluate the our NHHL algorithm. Finally, the simulation results demonstrate the superiority of the presented NHHL algorithm compared with the selected benchmark methods in various marine simulation scenarios.

Implicit Regularity and Linear Convergence Rates for the Generalized Trust-Region Subproblem

Implicit Regularity and Linear Convergence Rates for the Generalized Trust-Region Subproblem

Multistatic Localization by Differential Time Delays and Time Differences of Arrival in the Absence of Transmitter Position

We can only extract the differential time delay (DTD) measurements between the direct and reflected paths in multistatic localization when there is no synchronization in time between the transmitter and receiver and among the receivers. This paper first addresses the problem of multistatic localization of a fixed object when the transmitter position is not available by using the DTD measurements. We propose a two-step optimization method for jointly estimating the object and transmitter positions. In the first step, we formulate a non-convex constrained weighted least squares (CWLS) problem by transforming the DTD measurement model and introducing nuisance variables. Such a non-convex CWLS problem is then relaxed to a tractable convex semidefinite programming (SDP) problem by applying semidefinite relaxation. In the second step, the error coming from relaxation and approximation in the SDP solution is reduced iteratively through solving a generalized trust region subproblem (GTRS) in each iteration. If the receivers are synchronized such that the time difference of arrival (TDOA) measurements can be acquired in addition to DTD, we formulate a different CWLS problem by utilizing both DTD and TDOA measurements, which is solved by convex relaxation as well. The relaxed SDP problem can achieve the optimal solution of the CWLS problem, and further refinement is no longer needed. We conduct the mean square error (MSE) analysis to validate that both proposed methods are able to achieve the Cramer-Rao Lower Bound (CRLB) performance under small Gaussian noise, which is also validated by simulations.

Range Difference-Based Target Localization Under Stratification Effect and NLOS Bias in UWSNs

This letter proposes a robust target localization technique for acoustic underwater wireless sensor networks (UWSNs) that simultaneously experience the stratification effect and the non-line-of-sight (NLOS) bias. Specifically, we develop the range difference (RD)-based majorization minimization expression and its theoretical analysis, which leads to a surrogate function coupling with an iterative form. To figure out the initial estimate of the iteration, the proposed method converts the original localization problem into an RD-based generalized trust-region subproblem (RD-GTRS) after selecting a reference node. Eventually, an RD-based target localization approach (RD-TLA) is presented to iteratively estimate the final location. Simulation results show that the proposed RD-TLA method achieves better localization accuracy compared with other state-of-the-art methods in different scenarios.

Unknown Transmit Power RSSD-Based Localization in a Gaussian Mixture Channel

Localization is a critical task in wireless sensor networks (WSNs). However, accurate localization is difficult when there is noise from external sources and faulty nodes, especially when the transmit power is unknown. Signal strength-based localization methods are popular due to their low complexity and simple implementation. In this paper, robust localization based on the received signal strength difference (RSSD) is considered with unknown transmit power and Gaussian mixture noise in the presence of faulty nodes. A robust fault-tolerant localization (RFLT) technique is proposed by reformulating the original localization problem using a generalized trust-region subproblem (GTRS) framework. The resulting problem is non-convex so it is divided into two subproblems using a regularization item (RI) with a block-update surrogate function (BUSF). The active set method (ASM) is used to obtain an initial solution. The corresponding Cramer–Rao lower bound (CRLB) for Gaussian mixture noise is derived as a benchmark. Simulation results are presented which show that the performance of the proposed method is superior to other state-of-the-art methods.

Detecting Distance-Spoofing Attacks in Arbitrarily-Deployed Wireless Networks

The present work addresses localization problem in arbitrarily-deployed wireless networks in the presence of malicious attackers that can manipulate (spoof) their distance measurements. This problem is very important for the forthcoming applications that depend on accurate location information, since malicious attacks can potentially lead to disastrous consequences without the consideration of the security aspect. Therefore, this work proposes a new scheme for attacker detection based on generalized likelihood ratio test (GLRT) applied after a preliminary location estimate is obtained based on a simple geometric approach. The detected attacker is then removed from the localization process, which is converted into a generalized trust region sub-problem (GTRS) framework by applying the law of cosines (LC) and solved efficiently via bisectioning. The work also introduces theoretical bounds on the probability of detection in closed-form and studies the performance of the proposed algorithm from computational complexity, success in attacker detection, and localization accuracy perspectives. Our simulations corroborate the effectiveness of the proposed scheme, offering good localization accuracy and detection performance even in noisy environments

A Convex Optimization Approach For NLOS Error Mitigation in TOA-Based Localization

This paper addresses the target localization problem using time-of-arrival (TOA)-based technique under the non-line-of-sight (NLOS) environment. To alleviate the adverse effect of the NLOS error on localization, a total least square framework integrated with a regularization term (RTLS) is utilized, and with which the localization problem can get rid of the ill-posed issue. However, it is challenging to figure out the exact solution for the considered localization problem. In this case, we convert the RTLS problem into a semidefinite program (SDP), and then obtain the solution of the original problem by solving a generalized trust region subproblem (GTRS). The proposed method has a relatively good robustness in localization even under the circumstance that the prior knowledge of the NLOS links or its distribution does not know. The outperformance of the proposed method is demonstrated in the simulations compared with other state-of-the-art techniques.

Global and Asymptotically Efficient Localization From Range Measurements

We consider the range-based localization problem, which involves estimating an object's position by using $m$ sensors, hoping that as the number $m$ of sensors increases, the estimate converges to the true position with the minimum variance. We show that under some conditions on the sensor deployment and measurement noises, the LS estimator is strongly consistent and asymptotically normal. However, the LS problem is nonsmooth and nonconvex, and therefore hard to solve. We then devise realizable estimators that possess the same asymptotic properties as the LS one. These estimators are based on a two-step estimation architecture, which says that any $\sqrt{m}$-consistent estimate followed by a one-step Gauss-Newton iteration can yield a solution that possesses the same asymptotic property as the LS one. The keypoint of the two-step scheme is to construct a $\sqrt{m}$-consistent estimate in the first step. In terms of whether the variance of measurement noises is known or not, we propose the Bias-Eli estimator (which involves solving a generalized trust region subproblem) and the Noise-Est estimator (which is obtained by solving a convex problem), respectively. Both of them are proved to be $\sqrt{m}$-consistent. Moreover, we show that by discarding the constraints in the above two optimization problems, the resulting closed-form estimators (called Bias-Eli-Lin and Noise-Est-Lin) are also $\sqrt{m}$-consistent. Plenty of simulations verify the correctness of our theoretical claims, showing that the proposed two-step estimators can asymptotically achieve the Cramer-Rao lower bound.

Elliptic Localization With Imperfect Clock Synchronization for Known and Unknown Propagation Speed Scenarios

In this paper, we address the elliptic localization problem in the presence of clock synchronization errors at the receivers and transmitters, when the propagation speed is known as well as not known. For the known propagation speed scenario, we first show that the clock synchronization errors, when modeled as Gaussian random variables, can be treated as part of the composite noise in the measurement model without incurring performance loss in localization. Motivated by this result, we propose a generalized trust region subproblem solution for estimating the object position. For the unknown propagation speed scenario, we first formulate a non-convex weighted least squares problem. Such a non-convex problem is solved by either one of the two proposed methods, semidefinite relaxation or grid search, to jointly estimate the object position and the propagation speed. Furthermore, we conduct the mean square error analysis to show that the proposed methods are able to reach the Cramer-Rao lower bound accuracy under Gaussian noise, which is validated by using simulated and semi-real data.

Directional Target Localization in NLOS Environments Using RSS-TOA Combined Measurements

This letter addresses the localization problem of a directional source in non-line-of-sight (NLOS) environments. To estimate both the position and the orientation of the source, we propose an iterative algorithm that skillfully adopts the received-signal-strength (RSS) and time-of-arrival (TOA) measurements. The algorithm is derived by converting the non-convex estimating problem into a generalized trust region subproblem (GTRS). The converted problem could be efficiently solved by a bisection process. Considering that the directivity of the target could greatly affect its RSS measurements in different directions, a preliminary estimate for the target’s location is conducted using only the TOA measurements, based on which we then estimate the orientation of the source and the NLOS biases. The alternating process of the algorithm can converge quickly. Finally, the excellent performance of the proposed algorithm is demonstrated through simulations.

Maximum Correntropy Criterion for Robust TOA-Based Localization in NLOS Environments

We investigate the problem of time-of-arrival (TOA)-based localization under possible non-line-of-sight (NLOS) propagation conditions. To robustify the squared-range-based location estimator, we follow the maximum correntropy criterion, essentially the Welsch M-estimator with a redescending influence function which behaves like ell _0-minimization toward the grossly biased measurements, to derive the formulation. The half-quadratic technique is then applied to settle the resulting optimization problem in an alternating maximization (AM) manner. By construction, the major computational challenge at each AM iteration boils down to handling an easily solvable generalized trust region subproblem. It is worth noting that the implementation of our localization method requires nothing but merely the TOA-based range measurements and sensor positions as prior information. Simulation and experimental results demonstrate the competence of the presented scheme in outperforming several state-of-the-art approaches in terms of positioning accuracy, especially in scenarios, where the percentage of NLOS paths is not large enough.

A New Perspective on Range and Directional Localization Problem

This work studies the problem of target localization based on received signal strength (RSS) and angle of arrival (AOA) observations. In contrast to the existing solutions which base their derivations on separate manipulations of each of the measurement models and rely on additional quantity of information available through measurement fusion to boost their performance, we take a different approach here. Rather than treating the measurements individually, we combine them together in order to study the problem from a geometrical perspective. By exploiting the fact that the measurements are noise-corrupted, a fictitious sensor is introduced by each reference point in the network, which allows for application of law of cosines (LC) to fabricated triangles whose last vertex lies at the true target location. Based on weighted central mass of all fictitious sensors, additional side length of the triangles is estimated, which, together with the available angle information, enables straightforward conversion of the problem into a generalized trust region sub-problem (GTRS), which is solved exactly by a bisection method. Both cases of known and unknown transmit power are considered. The proposed algorithm has significantly inferior computational burden than the state-of-the-art approaches, and it not only matches their localization accuracy, but it outperforms them considerably for low number of reference points.

GTRS-Based Algorithm for UAV Navigation in Indoor Environments Employing Range Measurements and Odometry

This work addresses the problem of unmanned aerial vehicle (UAV) navigation in indoor environments. Due to unavailability of satellite signals, the proposed algorithm takes advantage of terrestrial radio measurements between the UAV and a set of stationary reference points, from which it extracts range information, as well as odometry by means of inertial sensors, such as accelerometer. On the one hand, based on maximum a posteriori (MAP) criterion, the range information and accumulated knowledge throughout the UAV's movement are employed to derive a generalized trust region sub-problem (GTRS), that is solved exactly via bisection procedure. On the other hand, by using the UAV's transform in relation to the world, another position estimation is obtained by employing odometry. Finally, the two position estimates are combined through a Kalman filter (KF) to enhance the positioning accuracy and obtain the final UAV's position estimation. The UAV is then navigated to a desired destination, by simply calculating the velocity components in the shortest path. Our results show that the proposed algorithm is robust to various model parameters for high precision (HP) UAV sensors, achieving reasonably good positioning accuracy. Besides, the results corroborate that the proposed algorithm is suitable for real-time applications, consuming (on average) only 21 ms to estimate the UAV position.

The generalized trust region subproblem: solution complexity and convex hull results

We consider the Generalized Trust Region Subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this problem recasts the GTRS as minimizing a linear objective subject to two nonconvex quadratic constraints. Our first main contribution is structural: we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil. This result may be of interest in building relaxations for nonconvex quadratic programs. Moreover, this result allows us to reformulate the GTRS as the minimization of two convex quadratic functions in the original space. Our next set of contributions is algorithmic: we present an algorithm for solving the GTRS up to an epsilon additive error based on this reformulation. We carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms. Notably, our algorithms run in linear (in the size of the input) time. Furthermore, our algorithm for computing an epsilon-optimal solution has a slightly-improved running time dependence on epsilon over the state-of-the-art algorithm. Our analysis shows that the dominant cost in solving the GTRS lies in solving a generalized eigenvalue problem -- establishing a natural connection between these problems. Finally, generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality-, interval-, and hollow- constrained variants of the GTRS. This gives the first linear-time algorithm in the literature for these variants of the GTRS.

RSS-Based Byzantine Fault-Tolerant Localization Algorithm Under NLOS Environment

Localization is one of the most critical tasks in wireless sensor networks, but achieving a relatively accurate location estimation is challenging when there have Byzantine fault and non-line-of-sight (NLOS) bias simultaneously. In this context, a localization method, based on received signal strength (RSS), is proposed in this letter to mitigate the impact of Byzantine fault and NLOS bias on the localization accuracy of wireless sensor networks. The proposed method relies on a Byzantine fault-tolerant localization algorithm (BFLA), which converts the localization problem into a generalized trust-region subproblem (GTRS) by applying certain approximations. In order to obtain a feasible solution to the GTRS, a block-coordinate update (BCU) function with a regularization term is used to divide the localization problem into two subproblems. An iterative method, whose start-point is obtained by an unconstrained squared-range (USR) algorithm, is then used to obtain a solution. Numerical simulations are carried out to show the effectiveness of the proposed method, compared with the state-of-the-art approaches in different scenarios.

An Absorption Mitigation Technique for Received Signal Strength-Based Target Localization in Underwater Wireless Sensor Networks.

Localization is an indispensable technology for underwater wireless sensor networks (UWSNs). In what concerns UWSNs, the accurate location information is not only the requirement of the marine field applications but also the basis of the other corresponding research, for instance, network routing and topology control. Recently, an astonishing surge of interest has been drawn in the received signal strength (RSS)-based scheme due to cost-effectiveness and synchronization-free compared with others. However, unlike the terrestrial wireless sensor networks (WSNs), the acoustic signal may suffer the absorption loss in the underwater environment besides the path loss, which degrades the localization accuracy and limits the capability of the RSS-based technology in UWSNs. In this context, a robust localization method with an absorption mitigation technique (AMT) is developed. First, an RSS-based analytically tractable measurement model is conducted, where the maximum likelihood estimator (MLE) is derived. Nevertheless, it is quite challenging to solve the problem using MLE under a non-convex expression. Therefore, by exploiting certain approximations, the considered localization problem is converted into an optimization expression with a maximum absorption loss involved. A min–max strategy is then presented, with which the problem is turned to minimize the worst situation of the absorption loss. After a simple manipulation, the problem is further investigated as a generalized trust region sub-problem (GTRS) framework. Although the GTRS is a non-convex scheme, the solution can be obtained through an iteration method by introducing a multiplier. In addition, the closed-form expression of the Cramer–Rao lower bound (CRLB) of the analytically tractable measurement model is derived. Numerical simulations demonstrate the effectiveness of the proposed method compared with the state-of-the-art approaches in different scenarios.

The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.

A Linear-Time Algorithm for Generalized Trust Region Subproblems

In this paper, we provide the first provable linear-time (in terms of the number of nonzero entries of the input) algorithm for approximately solving the generalized trust region subproblem (GTRS) of minimizing a quadratic function over a quadratic constraint under some regularity condition. Our algorithm is motivated by and extends a recent linear-time algorithm for the trust region subproblem by Hazan and Koren [Math. Program., 158 (2016), pp. 363--381]. However, due to the nonconvexity and noncompactness of the feasible region, such an extension is nontrivial. Our main contribution is to demonstrate that under some regularity condition, the optimal solution is in a compact and convex set and lower and upper bounds of the optimal value can be computed in linear time. Using these properties, we develop a linear-time algorithm for the GTRS.

A Geometric Approach for Distributed Multi-Hop Target Localization in Cooperative Networks

This work addresses target localization problem in cooperative distributed sensor networks, in which all sensors are capable of measuring Received Signal Strength (RSS), but only some are appropriately equipped to measure Angle Of Arrival (AOA) of the received signal. A novel approach based on simple geometry and multi-hopping is proposed, which allows for natural conversion of the problem into a Generalized Trust Region Sub-Problem (GTRS). The proposed algorithm comprises three main steps, each of them with linear computational cost in the number of neighbors, making it suitable for real-time applications. Our simulation results validate the performance of the new algorithm, surpassing some significantly more complex ones, and almost achieving a lower bound set by an existing algorithm which uses some (unrealistic) assumptions in its favor.