On the Non-global Local Minimizers of the Generalized Trust-region Subproblem and Its Equality-constrained Version: Number and Computation

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.

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.

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.

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.

Wireless localization is a fundamental component of modern sensor networks, with applications spanning environmental monitoring and smart cities. Ensuring accurate and efficient localization is critical for enhancing network performance and reliability, particularly in the presence of signal attenuation and noise. This study proposes a novel two-step localization framework, SD/SOCP-GTRS, to improve the precision of target localization using received signal strength (RSS) measurements. In the first step (SD/SOCP), semidefinite programming (SDP) and second-order cone programming (SOCP)-based convex relaxation are applied to the maximum likelihood (ML) estimator, generating an initial coarse estimate. The second step (GTRS) refines this estimate using weighted least squares (WLS) and the generalized trust region subproblem (GTRS), mitigating performance degradation caused by relaxation. Monte Carlo simulations validate that the proposed SD/SOCP-GTRS approach effectively reduces root mean square error (RMSE) compared to other methods. These findings demonstrate that the SD/SOCP-GTRS framework consistently outperforms existing techniques, approaching the theoretical performance limit and offering a robust solution for high-precision localization in wireless sensor networks.

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.

In this paper, we address the elliptic localization problem in the presence of clock synchronization errors at both the receivers and transmitters. To cope with this problem, we first show that the clock synchronization errors, modeled by Gaussian random variables, can be treated as a part of the composite noise without causing performance loss. Motivated by this finding, we propose a generalized trust region subproblem (GTRS) formulation for the elliptic localization problem where multiple transmitters are present. The global solution of the proposed GTRS is guaranteed although GTRS is a non-convex problem. Simulation results show that the proposed GTRS method outperforms the existing method.

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.

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 quadratic reformulation (CQR) via minimizing a linear objective over two convex quadratic constraints for the GTRS. We show that an optimal solution of the GTRS can be recovered from an optimal solution of the CQR. We further prove that this CQR is equivalent to minimizing the maximum of the two convex quadratic functions derived from the CQR for the case under investigation. Although the latter minimax problem is nonsmooth, it is well structured and convex. We thus develop two steepest descent algorithms corresponding to two different line search rules. We prove global sublinear convergence rates for both algorithms. We also obtain a local linear convergence rate of the first algorithm by estimating the Kurdyka-Łojasiewicz exponent at any optimal solution under mild conditions. We finally demonstrate the efficiency of our algorithms with numerical experiments.

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.

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.

Quadratically constrained quadratic programming is an important class of optimization problems. We consider the case with one quadratic constraint. Since both the objective function and its constraint can be neither convex nor concave, it is also known as the ‘generalized trust region subproblem.’ The theory and algorithms for this problem have been well studied under the Slater condition. In this article, we analyse the duality property between the primal problem and its Lagrangian dual problem, and discuss the attainability of the optimal primal solution without the Slater condition. The relations between the Lagrangian dual and semidefinite programming dual is also given.

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.

This work proposes a new algorithm to address the problem of unmanned aerial vehicle (UAV) navigation in satellite-less environments by combining machine learning with existent model-based methods. The proposed network model is trained by using the predictions of two estimators, one based on a Generalized trust region sub-problem (GTRS) framework and the other one founded on a Weighted Least Squares (WLS) principle. The solutions of these two estimators are then fed to two Long Short-Term Memories (LSTMs) to create models whose predictions are averaged to achieve the final prediction output. Our numerical results show favorable performance of the new network, obtaining improved accuracy and higher robustness to noise when compared with the individual counterparts of the network used in the training phase. Consequently, the proposed method offers safer an more reliable navigation of the UAV in satellite-less environments.

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.