Semidefinite Programming Method Research Articles

An Underwater Source Localization Method Using Bearing Measurements.

Angle-of-arrival (AOA) measurements are often used in underwater acoustical localization. Different from the traditional AOA model based on azimuth and elevation measurements, the AOA model studied in this paper uses bearing measurements. It is also often used in the Ultra-Short Baseline system (USBL). However, traditional acoustical localization needs additional range information. If the range information is unavailable, the closed-form solution is difficult to obtain only with bearing measurements. Thus, a localization closed-form solution using only bearing measurements is explored in this article. A pseudo-linear measurement model between the source position and the bearing measurements is derived, and considering the nonlinear relationship of the parameters, a weighted least-squares optimization equation based on multiple constraints is established. Different from the traditional two-step least-squares method, the semidefinite programming (SDP) method is designed to obtain the initial solution, and then a bias compensation method is proposed to further minimize localization errors based on the SDP result. Numerical simulations show that the performance of the proposed method can achieve Cramer-Rao lower bound (CRLB) accuracy. The field test also proves that the proposed method can locate the source position without range measurements and obtain the highest positioning accuracy.

Reach-Avoid Verification Based on Convex Optimization

In this paper we propose novel sufficient conditions for verifying reach-avoid properties of continuous-time systems modelled by ordinary differential equations (ODEs). Given a system, an initial set, a safe set and a target set of states, we say that the reach-avoid property holds, if for all initial conditions in the initial set, any trajectory of the system starting at them will eventually, i.e. in unbounded yet finite time, enter the target set while remaining inside the safe set until that first target hit (that is, if the system starting from the initial set can reach the target set safely). Based on a discount value function, two sets of quantified constraints are derived for verifying the reach-avoid property via the computation of exponential/asymptotic guidance-barrier functions (they form a barrier escorting the system to the target set safely at an exponential or asymptotic rate). It is interesting to find that one set of constraints whose solution is termed exponential guidance-barrier functions is just a simplified version of the existing one derived from the moment based method, while the other one whose solution is termed asymptotic guidance-barrier functions is completely new. Furthermore, built upon this new set of constraints, we derive a set of more expressive constraints, which includes the aforementioned two sets of constraints as special instances, providing more chances for verifying the reach-avoid property successfully. Finally, several examples demonstrate the theoretical developments and performance of proposed sufficient conditions using semi-definite programming methods.

Convex Combination for Wireless Localization Using Biased RSS Measurements

Received signal strength- (RSS-) based localization in wireless sensor networks (WSNs) has attracted significant attention due to its advantages of low cost and simple implementation. In practice, RSS measurements may suffer from sensor biases, which deteriorates the localization accuracy. However, most of the existing localization methods are designed for bias-free measurements. In this paper, we propose a convex combination method for RSS localization in the presence of sensor biases. The parameter vector composed of unknown location and sensor biases is estimated simultaneously by using a convex combination of some virtual points. These virtual points form a convex hull, into which the parameter vector falls with large probability. By this, the original nonconvex estimation problem is converted to be convex. Numerical examples demonstrate the superiority of the proposed method in terms of localization accuracy, compared to the existing semidefinite programming (SDP) methods.

Quantum-enhanced greedy combinatorial optimization solver.

Combinatorial optimization is a broadly attractive area for potential quantum advantage, but no quantum algorithm has yet made the leap. Noise in quantum hardware remains a challenge, and more sophisticated quantum-classical algorithms are required to bolster their performance. Here, we introduce an iterative quantum heuristic optimization algorithm to solve combinatorial optimization problems. The quantum algorithm reduces to a classical greedy algorithm in the presence of strong noise. We implement the quantum algorithm on a programmable superconducting quantum system using up to 72 qubits for solving paradigmatic Sherrington-Kirkpatrick Ising spin glass problems. We find the quantum algorithm systematically outperforms its classical greedy counterpart, signaling a quantum enhancement. Moreover, we observe an absolute performance comparable with a state-of-the-art semidefinite programming method. Classical simulations of the algorithm illustrate that a key challenge to reaching quantum advantage remains improving the quantum device characteristics.

IRS-Assisted Secure UAV Communication System for Multiuser With Hardware Impairments

Perfect hardware conditions are often difficult to meet in practical networks due to the inevitable hardware impairments (HIs). The secure multiuser communication of an intelligent reflecting surface (IRS)-aided unmanned aerial vehicle (UAV) system with HIs at the transceiver and the IRS is studied in this article. Aiming for maximizing the average sum secrecy rate (ASSR) of the UAV system, we jointly optimize the transmit precoding (TPC) matrices, IRS phase shift matrices, and UAV trajectory. To deal with the nonconvexity, we break down this problem into three subproblems and solve them using an alternating optimization (AO) algorithm. Specifically, the semidefinite programming method is used to solve the TPC and phase shift matrices, while the successive convex approximation method is used to obtain the UAV trajectory. In addition, a low complexity mathod on the basis of the Riemannian Manifold Gradient is proposed to solve the IRS phase shift matrix. Simulation results verify the effectiveness of the algorithm. Our robust algorithm can improve ASSR by 22% compared to that without IRS. Furthermore, the nonrobust design without considering HIs will cause a loss of secrecy performance.

Local convergence analysis of augmented Lagrangian method for nonlinear semidefinite programming

Local convergence analysis of augmented Lagrangian method for nonlinear semidefinite programming

Relative Transformation Estimation Based on Fusion of Odometry and UWB Ranging Data

In this article, we study the problem of estimating the four-degree-of-freedom (3-D position and heading) robot-to-robot relative frame transformation using onboard odometry and interrobot distance measurements. First, we present a theoretical analysis of the problem, namely, the derivation and interpretation of the Cramèr–Rao lower bound, the Fisher information matrix, and its determinant. Second, we propose optimization-based solutions, including a quadratically constrained quadratic programming (QCQP) formulation and its semidefinite programming (SDP) relaxation. Third, based on the theoretical results, we can detect singular configurations as well as measure the uncertainty of each individual parameter. We perform extensive simulations and real-life experiments with aerial robots to show that the proposed QCQP and SDP methods can outperform state-of-the-art approaches, especially in geometrically poor or large measurement noise conditions. In general, the QCQP method provides the best results at the expense of computational time, while the SDP method runs much faster and is sufficiently accurate in most cases.

Source Localization Using RSS Measurements with Sensor Position Uncertainty

Received signal strength- (RSS-) based localization has attracted considerable attention for its low cost and easy implementation. In plenty of existing work, sensor positions, which play an important role in source localization, are usually assumed perfectly known. Unfortunately, they are often subject to uncertainties, which directly leads to effect on localization result. To tackle this problem, we study the RSS-based source localization with sensor position uncertainty. Sensor position uncertainty will be modeled as two types: Gaussian random variable and unknown nonrandom variable. For either of the models, two semidefinite programming (SDP) methods are proposed, i.e., SDP-1 and SDP-2. The SDP-1 method proceeds from the nonconvex problem with respect to the maximum likelihood (ML) estimation and then obtains an SDP problem using proper approximation and relaxation. The SDP-2 method first transfers the sensor position uncertainties to the source position and then obtains an SDP formulation following a similar idea as in SDP-1 method. Numerical examples demonstrate the performance superiority of the proposed methods, compared to some existing methods assuming perfect sensor position information.

A preconditioned iterative interior point approach to the conic bundle subproblem

AbstractThe conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.

Federated learning via over-the-air computation in IRS-assisted UAV communications

Intelligent reflective surface (IRS) and unmanned aerial vehicle (UAV) communication are two key technologies in the sixth generation of mobile communication (6G). In this paper, IRS is equipped on UAV to form aerial IRS, which can achieve 360° panoramic full-angle reflection and flexible deployment of IRS. In order to achieve high-quality and ubiquitous network coverage under data privacy and low latency requirements, we propose an Federated learning (FL) network via Over-the-Air computation (AirComp) in IRS-assisted UAV communications. Our goal is to minimize the worst-case mean square error (MSE) by jointly optimizing the IRS phase shift, denoising factor for noise suppression, the user’s transmission power, and UAV trajectory. Optimizing and quickly adjusting the UAV position and IRS phase shift, it flexibly assists the signal transmission between users and base stations (BS). In order to solve this complex non-convex problem, we propose a low-complexity iterative algorithm, which divides the original problem into four sub-problems, respectively using the semi-definite programming (SDP) method, slack variable introduction method, successive convex approximation (SCA) method to solve each sub-problem. Through the analysis of simulation results, our proposed design scheme is obviously better than other benchmark schemes.

Average Rate Maximization for Mobile STAR-RIS-Aided NOMA System

Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) has been regarded as a revolutionary technology for the sixth generation (6G) mobile networks. This letter investigates the transmission scheme for mobile STAR-RIS-assisted non-orthogonal multiple access (NOMA) system. Specifically, an uplink average rate maximization problem is solved by jointly optimizing the phase shifts, the channel assignment, and the STAR-RIS’s location for mobile STAR-RIS-aided NOMA system. Particularly, a channel assignment scheme is developed to enhance the performance of NOMA system with multiple users. Furthermore, a successive convex approximation (SCA) method is invoked to address the STAR-RIS location design subproblem and a semi-definite programming (SDP) method is leveraged to solve the phase shift optimization subproblem. Simulation results show that the proposed mobile STAR-RIS-empowered NOMA method achieves better average rate performance than various benchmarking schemes.

UAV-Enabled Integrated Sensing, Computing, and Communication: A Fundamental Trade-Off

We propose an unmanned aerial vehicle (UAV) enabled integrated sensing, computing, and communication (U-ISCC) framework, where a treble-functional UAV performs local computation and also delivers its computational tasks to ground access points. Meanwhile, this UAV is employed to sense a target by using the unified beam waves. Our goal is to characterize the Pareto boundary between the computation capacity and the sensing beampattern gain, by which the fundamental trade-off of them is unveiled. To solve the challenging problem, we propose a joint optimization algorithm by applying the semidefinite programming method and the concave-convex procedure technique, which ensures the rank-1 constraint with a low complexity. Numerical results finally verify that the proposed algorithm explicitly depicts the performance region, and demonstrate that the UAV boosts the ability of sensing, computing, and communication to strike a balance by fully exploring its functions.

Joint Location and Beamforming Design for STAR-RIS Assisted NOMA Systems

Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) assisted non-orthogonal multiple access (NOMA) communication systems are investigated in its vicinity, where a STAR-RIS is deployed within a predefined region for establishing communication links for users. Both beamformer-based NOMA and cluster-based NOMA schemes are employed at the multi-antenna base station (BS). For each scheme, the STAR-RIS deployment location, the passive transmitting and reflecting beamforming (BF) of the STAR-RIS, and the active BF at the BS are jointly optimized for maximizing the weighted sum-rate (WSR) of users. To solve the resultant non-convex problems, an alternating optimization (AO) algorithm is proposed, where successive convex approximation (SCA) and semi-definite programming (SDP) methods are invoked for iteratively addressing the non-convexity of each sub-problem. Numerical results reveal that 1) the WSR performance can be significantly enhanced by optimizing the specific deployment location of the STAR-RIS; 2) both beamformer-based and cluster-based NOMA prefer asymmetric STAR-RIS deployment.

Robust Beamformer Design in Active RIS-Assisted Multiuser MIMO Cognitive Radio Networks

In this work, we address the problem of a joint robust transmission, reflection and reception strategy design at an active reconfigurable intelligent surface (RIS)-assisted underlay multiple input multiple output (MIMO) cognitive radio networks in which a secondary transmitter serves multiple secondary receivers simultaneously. As such, we study the impact of imperfection in channel state information (CSI) at the secondary transmitter following a norm-bounded error model. A sum mean squared error (MSE) minimization problem is formulated by jointly optimizing the transmit beamforming matrix (BFM) at the secondary transmitter, the linear reception filters (LRF) at the secondary receivers and the reflection coefficient matrix (RCM) at the RIS subject to the constraints of available transmission power at the secondary transmitter, maximum allowed interference power towards the primary receiver and the permissible amplification range at each RIS element. Due to the coupled nature of the BFM, LRF and RCM variables, the formulated problem is non-convex and, thus, cannot be solved using conventional optimization methods. As a result, we propose an alternating optimization (AO) based iterative algorithm that determines the optimal BFM, LRF and RCM using semidefinite programming and inner approximation methods with guaranteed convergence. We also discuss the impact of the limited power budget at active RIS on the performance of the considered network. We present simulation-based numerical results to validate the efficacy and robustness of the proposed algorithm. Moreover, we also highlight the superiority of the use of the active RIS over passive RIS in the considered communication network.

Efficient semidefinite solutions for TDOA-based source localization under unknown PS

Two efficient solutions via Semi-Definite Programming (SDP) are proposed for source localization problems using time difference of arrival (TDOA)-based ranging measurements when the propagation speed (PS) is unavailable and considered as a variable. For this problem, we propose a relaxed SDP (RSDP) solution, the performance of which is suboptimal. Accordingly, we propose a two-stage SDP method to improve the performance by applying the rank-reduction method. Besides, we also propose a penalty function-based SDP (PF-SDP) by introducing the penalty term. By doing so, the cost function becomes tighter so that the solution performs better. The simulated results show that the performance of two-stage SDP is sufficiently close to the Cramér-Rao Lower Bound (CRLB) accuracy at high noise levels. The PF-SDP outperforms the two-stage SDP in the presence of low noise levels.

Multi-User and Multi-Target Dual-Function Radar-Communication Waveform Design: Multi-Fold Performance Tradeoffs

The dual-functional radar-communication (DFRC) has played an essential role in the effective utilization of both hardware and spectrum resources. It enables the integrated sensing and communication (ISAC) towards 5G-Advanced or 6G. As an effective way to realize DFRC, in this paper, we design and optimize the transceiver waveform for a multi-input-multi-output (MIMO) DFRC system, where the dual functional base station (BS) simultaneously detects the multiple targets while communicating with multiple communication users. To this end, we first characterize the multi-fold tradeoffs between communication and radar sensing by defining an achievable performance region of the proposed DFRC system. Then, we formulate the optimal DFRC waveform design as a fairness profile optimization (FPO) problem to achieve the Pareto boundary of performance region, which is solved by using successive convex approximation (SCA) method. Moreover, two low-complexity waveform designs based on semidefinite programming and matrix space projection methods are provided to satisfy the goal of realizing green DFRC network. Simulation results confirm the superiority of our proposed designs.

An SDP method for fractional semi-infinite programming problems with SOS-convex polynomials

In this paper, we study a class of fractional semi-infinite polynomial programming problems involving sos-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules associated with the index set, a hierarchy of semidefinite programming (SDP) relaxations can be constructed and convergent upper bounds of the optimum can be obtained. In this paper, by introducing Lasserre’s measure-based representation of nonnegative polynomials on the index set to the conic reformulation, we present a new SDP relaxation method for the considered problem. This method enables us to compute convergent lower bounds of the optimum and extract approximate minimizers. Moreover, for a set defined by infinitely many sos-convex polynomial inequalities, we obtain a procedure to construct a convergent sequence of outer approximations which have semidefinite representations (SDr). The convergence rate of the lower bounds and outer SDr approximations are also discussed.

A revised sequential quadratic semidefinite programming method for nonlinear semidefinite optimization

In 2020, Yamakawa and Okuno proposed a stabilized sequential quadratic semidefinite programming (SQSDP) method for solving, in particular, degenerate nonlinear semidefinite optimization problems. The algorithm is shown to converge globally without a constraint qualification, and it has some nice properties, including the feasible subproblems, and their possible inexact computations. In particular, the convergence was established for approximate-Karush-Kuhn-Tucker (AKKT) and trace-AKKT conditions, which are two sequential optimality conditions for the nonlinear conic contexts. However, recently, complementarity-AKKT (CAKKT) conditions were also considered, as an alternative to the previous mentioned ones, that is more practical. Since few methods are shown to converge to CAKKT points, at least in conic optimization, and to complete the study associated to the SQSDP, here we propose a revised version of the method, maintaining the good properties. We modify the previous algorithm, prove the global convergence in the sense of CAKKT, and show some preliminary numerical experiments.

Recursive Convex Model for Optimal Power Flow Solution in Monopolar DC Networks

This paper presents a new optimal power flow (OPF) formulation for monopolar DC networks using a recursive convex representation. The hyperbolic relation between the voltages and power at each constant power terminal (generator or demand) is represented as a linear constraint for the demand nodes and generators. To reach the solution for the OPF problem a recursive evaluation of the model that determines the voltage variables at the iteration t+1 (vt+1) by using the information of the voltages at the iteration t (vt) is proposed. To finish the recursive solution process of the OPF problem via the convex relaxation, the difference between the voltage magnitudes in two consecutive iterations less than the predefined tolerance is considered as a stopping criterion. The numerical results in the 85-bus grid demonstrate that the proposed recursive convex model can solve the classical power flow problem in monopolar DC networks, and it also solves the OPF problem efficiently with a reduced convergence error when compared with semidefinite programming and combinatorial optimization methods. In addition, the proposed approach can deal with radial and meshed monopolar DC networks without modifications in its formulation. All the numerical implementations were in the MATLAB programming environment and the convex models were solved with the CVX and the Gurobi solver.

The Real Eigenpairs of Symmetric Tensors and Its Application to Independent Component Analysis.

It has been proved that the determination of independent components (ICs) in the independent component analysis (ICA) can be attributed to calculating the eigenpairs of high-order statistical tensors of the data. However, previous works can only obtain approximate solutions, which may affect the accuracy of the ICs. In addition, the number of ICs would need to be set manually. Recently, an algorithm based on semidefinite programming (SDP) has been proposed, which utilizes the first-order gradient information of the Lagrangian function and can obtain all the accurate real eigenpairs. In this article, for the first time, we introduce this into the ICA field, which tends to further improve the accuracy of the ICs. Note that the number of eigenpairs of symmetric tensors is usually larger than the number of ICs, indicating that the results directly obtained by SDP are redundant. Thus, in practice, it is necessary to introduce second-order derivative information to identify local extremum solutions. Therefore, originating from the SDP method, we present a new modified version, called modified SDP (MSDP), which incorporates the concept of the projected Hessian matrix into SDP and, thus, can intellectually exclude redundant ICs and select true ICs. Some cases that have been tested in the experiments demonstrate its effectiveness. Experiments on the image/sound blind separation and real multi/hyperspectral image also show its superiority in improving the accuracy of ICs and automatically determining the number of ICs. In addition, the results on hyperspectral simulation and real data also demonstrate that MSDP is also capable of dealing with cases, where the number of features is less than the number of ICs.