Solution Of Semidefinite Program Research Articles

Mitigating Sensor Motion Effect for AOA and AOA-TOA Localizations in Underwater Environments

The motion of sensors during the measurement period, if not accounted for, can degrade significantly the localization accuracy. This paper investigates the sensor motion effect for the positioning of an object, using angles of arrival only or together with time of arrival measurements. The biases from the AOA and TOA data models when ignoring the motion effect are examined. Positioning algorithms for AOA localization and mixed AOA-TOA localization that account for the motion effect are developed by the pseudo-linear formulation. The algorithms derived include the computationally attractive closed-form estimators and the noise resilient semidefinite programming solutions. The bias coming from the pseudo-linear formulation is analyzed in detail, and it can be subtracted from the closed-form solution to obtain a bias-suppressed estimate. Simulation validates the effectiveness of the proposed solutions in achieving the CRLB performance under Gaussian noise before the thresholding effect occurs.

Integrated sensing and communication waveform design in the Internet of Vehicles

We design an integrated sensing and communication (ISAC) transmission waveform with total transmission power, waveform similarity, and peak-to-average power ratio (PAPR) constraints in the Internet of Vehicles (IoV). The integrated waveform can complete radar target detection while communicating with downlink cellular users. The design of IoV with ISAC can not only improve the spectrum efficiency, but also further promote the integration of communication and sensing equipment. In this paper, the IoV with ISAC is formally modeled from the communication view and sensing view respectively. On this basis, taking the joint optimization of multi-user interference (MUI) total energy and Cramér Rao Bound (CRB) as the objective function, the transmission waveforms are designed with total transmission power, waveform similarity, and PAPR constraints respectively. To obtain the global optimal solution, we adopt a low complexity algorithm to solve the optimization problem under the total transmit power constraint and the optimization problem based on waveform similarity and PAPR constraints is further transformed into a nonconvex quadratically constrained quadratic programming (QCQP) problem, which is solved by using the semidefinite relaxation (SDR) technique. By ranking one approximation of the constrained optimization problem, it can be transformed into the solution of semidefinite programming (SDP) problems. The numerical results show that the designed ISAC transmission waveform can achieve an efficient tradeoff between sensing and communication performance, and each KPI is superior to one of the only communication and only sensing waveform design schemes. And it can improve the real-time information interaction ability of the IoV.

Moving Object Localization in Distributed MIMO With Clock and Frequency Offsets

This paper investigates the localization problem of the moving target in distributed multiple-input multiple-output (MIMO) systems, where the transmitters are subject to clock and frequency offsets. The position and velocity of the moving target are jointly determined by applying time delay (TD) and Doppler shift (DS) measurements. For this purpose, two efficient solution approaches are proposed to estimate the unknowns of the moving target, as well as the clock and frequency offsets. Firstly, the nonconvex localization problem is first relaxed into a convex semidefinite programming (SDP) form. Additionally, a two-stage closed-form solution (TSCFS) approach is proposed to determine the unknowns. The performance of the proposed methods is examined by the mean squared error (MSE) analysis. The simulated results show that the performance of the proposed solutions are able to approach the Cramér-rao lower bound (CRLB) accuracy at low noise levels. In contrast, the SDP solution performs better than the TSCFS in the presence of high noise levels or a small number of receivers.

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.

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.

Gridless DOA Estimation With Multiple Frequencies

Direction-of-arrival (DOA) estimation is widely applied in acoustic source localization. A multi-frequency model is suitable for characterizing the broadband structure in acoustic signals. In this paper, the continuous (gridless) DOA estimation problem with multiple frequencies is considered. This problem is formulated as an atomic norm minimization (ANM) problem. The ANM problem is equivalent to a semi-definite program (SDP) which can be solved by an off-the-shelf SDP solver. The dual certificate condition is provided to certify the optimality of the SDP solution so that the sources can be localized by finding the roots of a polynomial. We also construct the dual polynomial to satisfy the dual certificate condition and show that such a construction exists when the source amplitude has a uniform magnitude. In multi-frequency ANM, spatial aliasing of DOAs at higher frequencies can cause challenges. We discuss this issue extensively and propose a robust solution to combat aliasing. Numerical results support our theoretical findings and demonstrate the effectiveness of the proposed method.

Semidefinite Programming Solutions for Elliptic Localization in Asynchronous Radar Networks

In this article, semidefinite programming (SDP) solutions are proposed for the elliptic localization problem in asynchronous radar networks, where the transmitters are subject to clock offsets. Relaxing the nonconvex problem into different convex forms, we design tight SDP (TSDP) and global SDP (GSDP) solutions for this problem. The TSDP solution includes several SDP cones to solve the problem, and the correlation is not considered. Hence, the TSDP solution performs poorly in the presence of correlated noise. We further put forward the GSDP solution to improve the performance by introducing a new GSDP form, which includes only one SDP cone to handle the correlated noise. We also theoretically prove that the GSDP problem is tight enough so that its performance is able to sufficiently approach the Cramér–Rao lower bound (CRLB) accuracy. The simulated results show that the TSDP solution can provide the comparable performance with the GSDP in the absence of related noise. The performance of the GSDP solution can almost attain the CRLB accuracy using less receivers.

Efficient solutions for target localization in asynchronous MIMO networks

Multiple-input multiple-output (MIMO) technique has been drawing extensive attention for its spatial diversity to enhance the performance in the target localization. Considering a fully asynchronous MIMO network, we propose a differential delay time (DDT) method for target localization. Semidefinite programming (SDP) and closed-form solutions (CFS) always converge to global optimum. To efficiently solve the DDT-based localization problem, we propose a global SDP (GSDP) solution to determine the target position. The complexity of the GSDP solution is high. Hence, we also develop a two-stage CFS (TSCFS) to reduce the complexity. We also prove that the performance of the TSCFS and GSDP solutions is able to attain the Cramér–Rao Lower Bound (CRLB) accuracy. The simulated results show that the GSDP solution provides comparable performance with the TSCFS at the low noise levels. Compared with the TSCFS, the GSDP shows its superiority in the estimation performance at the high noise levels.

Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

In this paper, we propose a primal-dual path following method for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs. In contrast, our approach does not rely on solving SDPs accurately but approximately following a path leading to a solution, which is formed on the intersection of the semi-infinite feasible region and the interior of the semi-definite feasible region. Specifically, we first present a prototype path-following method and show its global weak* convergence to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions. Next, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with the Monteiro-Zhang scaling technique into the prototype method. We prove two-step superlinear convergence of the resulting algorithm using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP.

Hidden Integrality and Semirandom Robustness of SDP Relaxation for Sub-Gaussian Mixture Model

We consider the problem of estimating the discrete clustering structures under the sub-Gaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “global-to-local” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of [Formula: see text] fraction of the points for any [Formula: see text]. As a special case, we show that under the [Formula: see text]-dimensional stochastic ball model, SDP achieves nontrivial (sometimes exact) recovery when the center separation is as small as [Formula: see text], which improves upon previous exact recovery results that require constant separation.

Exploiting Reconfigurable Intelligent Surfaces in Edge Caching: Joint Hybrid Beamforming and Content Placement Optimization

Edge caching can effectively reduce backhaul burden at core network and increase quality-of-service at wireless edge nodes. However, the beneficial role of edge caching cannot be fully realized when the offloading link is in deep fade. Fortunately, the impairments induced by wireless propagation environments could be renovated by a reconfigurable intelligent surface (RIS). In this paper, a new RIS-aided edge caching system is proposed, where a network cost minimization problem is formulated to optimize content placement at cache units, active beamforming at base station and passive phase shifting at RIS. After decoupling the content placement subproblem with the hybrid beamforming design, we propose an alternating optimization algorithm to tackle the active beamforming and passive phase shifting. For active beamforming, we transfer the problem into a semidefinite programming (SDP) and prove that the optimal solution of SDP is always rank-one. For passive phase shifting, we introduce the block coordinate descent method to alternately optimize the auxiliary variables and the RIS phase shifts. Further, a conjugate gradient algorithm based on manifold optimization is proposed to deal with the non-convex unit-modulus constraints. Numerical results show that our RIS-aided edge caching design can effectively decrease the network cost by improving the quality of offloading links.

User Selection for MIMO Downlink With Digital and Hybrid Maximum Ratio Transmission

In this paper, we propose a user selection algorithm suitable for digital and hybrid maximum ratio transmission (MRT) precoding to alleviate inter-user interference. For digital MRT, the minimum inter-user interference problem is reduced to the notorious NP-hard called max k -cut. We first relax the max k -cut problem into a semi-definite programming (SDP), which can be solved by the interior point method. Then, we propose a randomized rounding-based greedy user selection algorithm (RRGUSA) to get the approximate solution of the max k -cut from the SDP solution. For hybrid MRT, the minimum inter-user interference problem is more complicated. We first force the interference minimization problem of hybrid MRT to be a quasi-max k -cut problem according to the law of large numbers. Then, RRGUSA is still used to obtain an approximate solution. Simulation results show that, for digital MRT, the proposed algorithm obtains a near-optimal solution, and for hybrid MRT, the proposed algorithm brings better performance by deploying more antennas.

Semidefinite Programming for Community Detection With Side Information

This paper produces an efficient semidefinite programming (SDP) solution for community detection that incorporates non-graph data, which in this context is known as side information. SDP is an efficient solution for standard community detection on graphs. We formulate a semi-definite relaxation for the maximum likelihood estimation of node labels, subject to observing both graph and non-graph data. This formulation is distinct from the SDP solution of standard community detection, but maintains its desirable properties. We calculate the exact recovery threshold for three types of non-graph information, which in this paper are called side information: partially revealed labels, noisy labels, as well as multiple observations (features) per node with arbitrary but finite cardinality. We find that SDP has the same exact recovery threshold in the presence of side information as maximum likelihood with side information. Thus, the methods developed herein are computationally efficient as well as asymptotically accurate for the solution of community detection in the presence of side information. Simulations show that the asymptotic results of this paper can also shed light on the performance of SDP for graphs of modest size.

Rank-One Semidefinite Programming Solutions for Mobile Source Localization in Sensor Networks

In this paper, we introduce a rank-one Semi-Definite Programming (SDP) solution method for mobile source localization in sensor networks. The position and velocity of mobile source are jointly estimated using Time Delay (TD) measurements. To obtain the position and velocity of mobile source, a Relaxed Semi-Definite Programming (RSDP) algorithm is firstly designed by dropping the rank-one constraint. However, dropping the rank-one constraint leads to produce a suboptimal solution. To improve the performance, we further put forward the Penalty Function Semi-Definite Programming (PF-SDP) method to obtain the rank-one solution of the estimation problem by introducing the penalty terms. By adaptively choosing the penalty coefficient, an Adaptive Penalty Function Semi-Definite Programming (APF-SDP) algorithm is also proposed to avoid the excessive penalty. We also conduct experiments in both a simulated environment and a real system to demonstrate the effectiveness of the proposed methods. The results have demonstrated that the proposed APF-SDP outperforms the PF-SDP in terms of the position and velocity estimation whether the noise level is large or not.

Quantum algorithms and lower bounds for convex optimization

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an n-dimensional convex body using O~(n) queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires Ω~(n) evaluation queries and Ω(n) membership queries.

A mixed-integer SDP solution approach to distributionally robust unit commitment with second order moment constraints

A power system unit commitment (UC) problem considering uncertainties of renewable energy sources is investigated in this paper, through a distributionally robust optimization approach. We assume that the first and second order moments of stochastic parameters can be inferred from historical data, and then employed to model the set of probability distributions. The resulting problem is a two-stage distributionally robust unit commitment with second order moment constraints, and we show that it can be recast as a mixed-integer semidefinite programming (MI-SDP) with finite constraints. The solution algorithm of the problem comprises solving a series of relaxed MI-SDPs and a subroutine of feasibility checking and vertex generation. Based on the verification of strong duality of the semidefinite programming (SDP) problems, we propose a cutting plane algorithm for solving the MI-SDPs; we also introduce a SDP relaxation for the feasibility checking problem, which is an intractable biconvex optimization. Experimental results on a IEEE 6-bus system are presented, showing that without any tunings of parameters, the real-time operation cost of distributionally robust UC method outperforms those of deterministic UC and two-stage robust UC methods in general, and our method also enjoys higher reliability of dispatch operation.

Range-Based Rigid Body Localization With a Calibration Emitter for Mitigating Anchor Position Uncertainties

Rigid body localization (RBL) extends the traditional point positioning by determining not only the position but also the orientation of the body. This paper considers RBL using the range measurements between the sensors on the body and the outside anchors that have position uncertainties, where a calibration emitter at an inaccurate location is employed to mitigate the anchor position errors. This is a highly nonlinear constrained optimization problem as the rotation matrix defining the orientation must belong to the special orthogonal group. We first propose the use of rotation angles to parameterize the orientation, which enables the reduction of the problem to an unconstrained optimization. It is then sufficient to consider the unconstrained Cramer-Rao Lower Bound (CRLB) for analyzing the effects of anchor position errors and noisy calibration position on the localization performance, and such an analysis is prohibitive when using the constrained CRLB. We next propose and analyze three Maximum Likelihood estimators obtained from the unconstrained formulation that have different levels of approximation and complexity. We also enhance the divide and conquer and the semi-definite programming solution from the literature such that they can work with the presence of a calibration emitter and account for anchor position errors.

Localization Using Blind RSS Measurements

Localization using received signal strength (RSS) measurements becomes popular due to the simplicity of practical implementation. Traditional RSS measurements are obtained after successful demodulation such that the impact of the background noise (BGN) is ignored. However, critical information for demodulation might be expensive or difficult to obtain in hostile or harsh environments. In this case, the RSS measurements need to be blindly collected without demodulation and hence characterized by a recent model with the BGN power (already validated by real-life data). This kind of measurement is referred to as “blind RSS measurement”. In this letter, we introduce four models for the localization using the blind RSS measurements, respectively considering the BGN power and the transmit power to be known or unknown. A general semi-definite programming solution that applies to all these models is proposed. The corresponding Cramer–Rao lower bounds are presented, indicating a significant impact of the BGN power on the estimation accuracy. Numerical results show the proposed method yields a good and reliable performance with different models.

Detection of shockable ventricular arrhythmia using optimal orthogonal wavelet filters

Sudden cardiac death (SCD) is caused by lethal arrhythmia. Ventricular fibrillation (VF) and ventricular tachycardia (VT) are amenable to defibrillation or electrical shock therapy (“shockable” arrhythmia) that can abolish the VF/VT and restore normal electrical and mechanical heart function. The challenge is to differentiate between shockable and non-shockable arrhythmia during the emergency response to SCD. When it comes to saving the life, accurate electrocardiogram (ECG) diagnosis and fast delivery of appropriate treatment is imperative. Automated systems to differentiate shockable from non-shockable arrhythmia have been developed to overcome the difficulty, and possible errors due to the manual inspection. In the present work, we have devised an efficient, effective and robust automated system to detect shockable and non-shockable arrhythmia using an optimal wavelet-based features extracted from ECG epochs of 2 s durations. We employed optimal two-channel frequency selective orthogonal wavelet filter bank to diagnose shockable ventricular arrhythmia. The optimization was carried out by minimizing the stop band ripple energy of the wavelet filter. The optimal orthogonal wavelet filter has been designed using a semi-definite programming (SDP) formulation without the use of any parameterization. The SDP solution gave us the desired optimal orthogonal wavelet filter bank with minimum stop band energy and the desired degree of regularity for the given length of filter. Fuzzy entropy and Renyi entropy features were extracted from the 2-s ECG epochs. These extracted features were then fed into the classifiers for discrimination of shockable arrhythmia rhythms and non-shockable arrhythmia rhythms. The best results were obtained from support vector machine. Accuracy of 97.8%, sensitivity of 93.42%, and specificity of 98.35% were obtained using a tenfold cross validation scheme. The developed automated system is accurate and robust; therefore, it can be integrated in automated external defibrillators that can be deployed for hospitals as well as out-of-hospital emergency resuscitation of SCD.

Exponential Error Rates of SDP for Block Models: Beyond Grothendieck’s Inequality

In this paper, we consider the cluster estimation problem under the stochastic block model. We show that the semidefinite programming (SDP) formulation for this problem achieves an error rate that decays exponentially in the signal-to-noise ratio. The error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime. An immediate corollary of our results yields error bounds under the censored block model. Moreover, these error bounds are robust, continuing to hold under heterogeneous edge probabilities and a form of the so-called monotone attack. Significantly, this error rate is achieved by the SDP solution itself without any further pre- or post-processing and improves upon existing polynomially decaying error bounds proved using the Grothendieck’s inequality. Our analysis builds on two key ingredients: 1) showing that the graph has a well-behaved spectrum, even in the sparse regime, after discounting an exponentially small number of edges and 2) an order-statistics argument that governs the final error rate. Both arguments highlight the implicit regularization effect of the SDP formulation.