Convex Semidefinite Programming Problem Research Articles

Characterizations of ε-Approximate Solutions for Robust Convex Semidefinite Programming Problems

Characterizations of ε-Approximate Solutions for Robust Convex Semidefinite Programming Problems

Optimal linear array orientation design for 3D direct position determination via semi-Definite relaxation

This paper presents an optimization strategy for array orientations in a three-dimensional (3D) direct position determination (DPD) system. Specifically, we consider a scenario in which uniform linear arrays (ULAs) are used to locate emitters, and we seek to optimize the array orientations in terms of localization accuracy. The E-optimality criterion, which minimizes the spectral norm of the Cramér-Rao lower bound (CRLB), is exploited to formulate this problem. As the objective function is non-convex with bilinear structures, we leverage semi-definite relaxation (SDR) to transform it into a convex semi-definite programming (SDP) problem by substituting the bilinear terms with a matrix variable. In addition, we tackle the array orientation configuration problem in the context of multi-emitter scenarios and develop an SDR solution for large-scale antenna array systems. Simulation results validate that the ULAs with array orientations designed by the proposed SDR-based method have near-optimal localization performance.

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.

Convex relaxation for optimal fixture layout design

This article proposes a general fixture layout design framework that directly integrates the system equation with the convex relaxation method. Note that the optimal fixture design problem is a large-scale combinatorial optimization problem; we relax it to a convex Semi-Definite Programming (SDP) problem by adopting sparse learning and SDP relaxation techniques. It can be solved efficiently by existing convex optimization algorithms and thus generates a near-optimal fixture layout. A real case study in the half-to-half fuselage assembly process indicates the superiority of our proposed algorithm compared to the current industry practice and state-of-art methods.

Multistatic Localization With Unknown Transmitter Position and Signal Propagation Speed

In this letter, we address the multistatic object localization problem when the transmitter position and signal propagation speed are not known. By transforming the time delay measurement model, we first formulate a non-convex weighted least squares problem, which is then relaxed into a convex semidefinite programming (SDP) problem by applying semidefinite relaxation. It turns out that the relaxed SDP problem is too loose and fails to achieve the Cramer-Rao lower bound (CRLB) performance. To address this problem, we propose to tighten the relaxed SDP problem by adding a second-order cone constraint to reach better solutions. Simulation results show that the proposed method is able to achieve the CRLB.

3-D Target Localization Based on Bi-static Range Measurements in Widely Separated MIMO Radars

In this paper, a three-dimensional target localization problem in widely separated multiple-input multiple-output radars is solved using two specific techniques based on time difference of arrival measurements. These techniques are provided in terms of transmitter and receiver antennas, which are named as technique_t and technique_r, respectively. The localization problem is rewritten as a non-convex optimization problem which is based on a least-squares method without any initial estimation. Therefore, a convex semidefinite programming problem is obtained by utilizing the semidefinite relaxation method for the problem which can be performed via the CVX toolbox. Several simulations are provided to evaluate the positioning accuracy in terms of bi-static range error for 3 and 4 transmitter/receiver antennas, different antenna arrangements, and near/far target. In other simulations, the localization accuracy is evaluated in terms of the empirical cumulative density function of positioning error. The results show that the proposed techniques have better accuracy and performance in different scenarios in comparison with other compared methods. The last simulation also demonstrates that the computational time of the mentioned techniques is 0.69 s which is suitable for real-time processing.

Multi-Party Dynamic State Estimation That Preserves Data and Model Privacy

In this paper we focus on the dynamic state estimation which harnesses a vast amount of sensing data harvested by multiple parties and recognize that in many applications, to improve collaborations between parties, the estimation procedure must be designed with the awareness of protecting participants’ data and model privacy, where the latter refers to the privacy of key parameters of observation models. We develop a state estimation paradigm for the scenario where multiple parties with data and model privacy concerns are involved. Multiple parties monitor a physical dynamic process by deploying their own sensor networks and update the state estimate according to the average state estimate of all the parties calculated by a cloud server and security module. The paradigm taps additively homomorphic encryption which enables the cloud server and security module to jointly fuse parties’ data while preserving the data privacy. Meanwhile, all the parties collaboratively develop a stable (or optimal) fusion rule without divulging sensitive model information. For the proposed filtering paradigm, we analyze the stabilization and the optimality. First, to stabilize the multi-party state estimator while preserving observation model privacy, two stabilization design methods are proposed. For special scenarios, the parties directly design their estimator gains by the matrix norm relaxation. For general scenarios, after transforming the original design problem into a convex semi-definite programming problem, the parties collaboratively derive suitable estimator gains based on the alternating direction method of multipliers (ADMM). Second, an optimal collaborative gain design method with model privacy guarantees is provided, which results in the asymptotic minimum mean square error (MMSE) state estimation. Finally, numerical examples are presented to illustrate our design and theoretical findings.

Pairs of Pilots Design for Preamble-Based Channel Estimation in OQAM/FBMC Systems

In this letter, we propose an improved pairs of pilots (POP) design method for channel estimation in offset quadrature amplitude modulation based filter bank multicarrier (OQAM/FBMC) systems. Based on an error analysis of the conventional POP channel estimation method, we first develop a heuristic pilot design in which each pilot pair is inserted at two consecutive subcarriers. Besides, we further propose to build a preamble by quartet of subcarriers and conduct a pilot optimization subject to a pilot power constraint, aiming to suppress the noise effect as much as possible. To obtain its optimal solution, we convert the original non-convex pilot optimization problem into a convex semidefinite programming problem, which is shown to be computationally efficient and readily solvable. Numerical simulations validate the effectiveness and superiority of the proposed pilot design method compared to the conventional POP and interference approximation methods.

Secure transmission for wireless collaborative systems based on second‐order channel statistics

AbstractIn this article, we investigate the maximum secrecy rates (SRs) and the optimal beamforming schemes for a collaborative wireless network under uncertain channel levels of available second‐order statistics. Both decode‐and‐forward (DF) and amplify‐and‐forward (AF) strategies under the aggregate and individual power constraints are considered. We have derived the maximum SRs and the optimal beamforming vectors under aforementioned conditions and an efficient cooperation paradigm to provide secrecy in a multirelay environment. A closed‐form maximum SR in the DF scheme under aggregate power constraint is presented. Next, the upper and lower bounds for the maximum SR under aggregate power constraint in the AF scheme is derived. The optimization problem in the AF scheme can be converted into a convex semidefinite programming problem, which is efficiently solved by the bisection method. Moreover, the optimal beamforming weights are also obtained to maximize the SRs under both aggregate and individual relay power constraints. Numerical results are provided to demonstrate the effectiveness of proposed algorithms.

Modal dynamic residual-based model updating through regularized semidefinite programming with facial reduction

Structural model updating techniques optimize model parameter values for improving the predication accuracy of a numerical model. Formulated as optimization problems, the objective of these techniques is to minimize the difference between model prediction and the experimental data. Most of these optimization problems are nonconvex, and consequently, the global optimum is very hard to solve. The sum of squares (SOS) approach and its sparse variant have been reported to relax a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem, which is more readily solvable. However, the corresponding convex SDP problem may fail the Slater condition qualification. The failure increases the difficulty for numerical algorithms, e.g. the interior point method, to solve the SDP problem. This paper proposes to utilize the facial reduction technique to regularize an SDP problem which fails the Slater condition qualification. In addition, the regularized SDP problem has smaller size, which helps to improve the computation efficiency. The performance of the proposed model updating approach is evaluated through a plane truss example.

Least Squared Relative Error Estimator for RSS Based Localization With Unknown Transmit Power

Research on received signal strength (RSS) based localization has received a lot of attention from both academia and industry because of its low complexity and high efficiency. In this letter, the RSS based localization problem when the transmit power is unknown, is addressed based on the least squared relative error (LSRE) estimation. By taking exponential transformation, the original log-normal RSS measurement model is transformed into a multiplicative model, which is used to formulate a non-convex LSRE estimation problem with the source location and the transmit power as variables. We then apply semidefinite relaxation (SDR) to the non-convex LSRE problem to obtain a convex semidefinite programming (SDP) problem. To facilitate SDR, we introduce two compound variables constructed by the source location and the transmit power, and then estimate the two compound variables instead of directly estimating the source location and the transmit power. The source location estimate is recovered according to its relation with the compound variable. Both simulations and real field test demonstrate the superior performance of the proposed method over several existing methods.

Quadratic Matrix Inequality Approach to Robust Adaptive Beamforming for General-Rank Signal Model

The worst-case robust adaptive beamforming problem for general-rank signal model is considered. This is a nonconvex problem, and an approximate version of it (obtained by introducing a matrix decomposition on the presumed covariance matrix of the desired signal) has been well studied in the literature. Different from the existing literature, herein however the original beamforming problem is tackled. Resorting to the strong duality of linear conic programming, the robust adaptive beamforming problem for general-rank signal model is reformulated into an equivalent quadratic matrix inequality (QMI) problem. By employing a linear matrix inequality (LMI) relaxation technique, the QMI problem is turned into a convex semidefinite programming problem. Using the fact that there is often a positive gap between the QMI problem and its LMI relaxation, an approximate algorithm is proposed to solve the robust adaptive beamforming in the QMI form. Besides, several sufficient optimality conditions for the nonconvex QMI problem are built. To validate our results, simulation examples are presented, which also demonstrate the improved performance of the new robust beamformer in terms of the output signal-to-interference-plus-noise ratio.

Usage of SDP Relaxation in Some GNSS Navigation Problems

In this paper, we propose an approach to the solution of two optimization problems that arise when solving problems of satellite navigation. The first problem consists in attitude determination using carrier phase and code range observables measured by multiple GNSS antennas. In general, the attitude determination is the non-convex optimization problem. The second problem consists in the optimization of the set of the Global Navigation Satellite System (GNSS) signals chosen for processing when solving the positioning problem. It is the binary optimization problem which is hard for a solution in real time. We show how these problems can be reduced to the convex semidefinite programming problem (SDP) and, therefore, efficiently solved.

Sparse Sum-of-Squares Optimization for Model Updating Through Minimization of Modal Dynamic Residuals

This research investigates the application of sum-of-squares (SOS) optimization method on finite element model updating through minimization of modal dynamic residuals. The modal dynamic residual formulation usually leads to a nonconvex polynomial optimization problem, the global optimality of which cannot be guaranteed by most off-the-shelf optimization solvers. The SOS optimization method can recast a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem. However, the size of the SDP problem can grow very large, sometimes with hundreds of thousands of variables. To improve the computation efficiency, this study exploits the sparsity in SOS optimization to significantly reduce the size of the SDP problem. A numerical example is provided to validate the proposed method.

Bayesian Compressive Sensing Based Optimized Node Selection Scheme in Underwater Sensor Networks

Information acquisition in underwater sensor networks is usually limited by energy and bandwidth. Fortunately, the received signal can be represented sparsely on some basis. Therefore, a compressed sensing method can be used to collect the information by selecting a subset of the total sensor nodes. The conventional compressed sensing scheme is to select some sensor nodes randomly. The network lifetime and the correlation of sensor nodes are not considered. Therefore, it is significant to adjust the sensor node selection scheme according to these factors for the superior performance. In this paper, an optimized sensor node selection scheme is given based on Bayesian estimation theory. The advantage of Bayesian estimation is to give the closed-form expression of posterior density function and error covariance matrix. The proposed optimization problem first aims at minimizing the mean square error (MSE) of Bayesian estimation based on a given error covariance matrix. Then, the non-convex optimization problem is transformed as a convex semidefinite programming problem by relaxing the constraints. Finally, the residual energy of each sensor node is taken into account as a constraint in the optimization problem. Simulation results demonstrate that the proposed scheme has better performance than a conventional compressed sensing scheme.

Robust Beamforming for Physical Layer Security in BDMA Massive MIMO

In this paper, we design robust beamforming to guarantee the physical layer security for a multiuser beam division multiple access (BDMA) massive multiple-input multiple-output (MIMO) system, when the channel estimation errors are taken into consideration. With the aid of artificial noise, the proposed design are formulated as minimizing the transmit power of the base station, while providing legal users and the eavesdropper with different signal-to-interference-plus-noise ratio. It is strictly proved that, under BDMA massive MIMO scheme, the initial non-convex optimization can be equivalently converted to a convex semi-definite programming problem and the optimal rank-one beamforming solutions can be guaranteed. In stead of directly resorting to the convex tool, we make one step further by deriving the optimal beamforming direction and the optimal beamforming power allocation in closed-form, which greatly reduces the computational complexity and makes the proposed design practical for real world applications. Simulation results are then provided to verify the efficiency of the proposed algorithm.

Robust Downlink Beamforming for BDMA Massive MIMO System

In this paper, we design robust downlink beamforming against the imperfect channel state information (CSI) for beam division multiple access (BDMA) massive multiple-input multiple output (MIMO) systems. Following a worst-case deterministic model, the proposed design is formulated as minimizing the power consumption of base station (BS) under different signal-to-interference-plus-noise ratio (SINR) constraints. The S-Procedure and semi-definite relaxation (SDR) are used to convert the initial non-convex optimization to a convex semi-definite programming problem. Then the optimality of SDR is strictly proved by showing the rank-one property of the optimal beamforming thanks to the orthogonal channels under BDMA scheme. More importantly, we make one step further by deriving the optimal beamforming directions and optimal beamforming power allocation of the SDR in closed-form, which greatly reduces the optimization complexity and makes the proposed design practical for a real word massive MIMO system. Simulation results are then provided to verify the efficiency of the proposed robust beamforming algorithm.

Doppler Frequency Shift Based Source Localization in Presence of Sensor Location Errors

In this paper, the Doppler frequency shift-based localization problem in the presence of sensor location errors is addressed. Based on the measurement model, we present two methods, an explicit solution and a semidefinite relaxation (SDR) technique, to estimate the position of the stationary source. In the two proposed algorithms, the non-accurate information on sensor position and velocity is considered. In the former algorithm, based on two best linear unbiased estimator, an explicit solution of the source position can be obtained, whereas in the latter one, the SDR technique is applied to relax the original maximum likelihood estimation into a convex semidefinite programming problem, which provides accurate estimate without postprocessing. Compared with the methods where the sensor location errors are not taken into consideration, the proposed algorithms are able to achieve a more accurate localization result. Simulations verify a better performance of these two proposed algorithms.

On approximate solutions for robust convex semidefinite optimization problems

In this paper, we consider approximate solutions ( $$\epsilon $$ -solutions) for a convex semidefinite programming problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we prove an approximate optimality theorem and approximate duality theorems for $$\epsilon $$ -solutions in robust convex semidefinite programming problem under the robust characteristic cone constraint qualification. Moreover, an example is given to illustrate the obtained results.

Approximate Optimality and Approximate Duality for Quasi Approximate Solutions in Robust Convex Semidefinite Programs

In this paper, we study quasi approximate solutions for a convex semidefinite programming problem in the face of data uncertainty. Using the robust optimization approach (worst-case approach), approximate optimality conditions and approximate duality theorems for quasi approximate solutions in robust convex semidefinite programming problems are explored under the robust characteristic cone constraint qualification. Moreover, some examples are given to illustrate the obtained results.