Convex Semidefinite Programming Research Articles

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.

Computation of Invariant Tubes for Robust Output Feedback Model Predictive Control

This paper presents an algorithm to calculate tightened invariant tubes for output feedback model predictive controllers (MPC). We consider discrete-time linear time-invariant (DLTI) systems with bounded state and input constraints and subject to bounded disturbances. In contrast to existing approaches which either use pre-defined control and observer gains or compute the control and observer gains that optimize the volume of the invariant sets for the estimation and control errors separately, we consider the problem of optimizing the volume of these sets simultaneously. The nonlinearities associated with computing the control and observer gains are circumvented by the application of Farkas’ Theorem and an extended Elimination Lemma, to convert the nonconvex optimization problem into a convex semidefinite program. An update algorithm is then used to reduce the volume of the invariant tube through a finite number of iterations. Numerical examples are provided to illustrate the effectiveness of the proposed algorithm.

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.

Schedule Reliability in Liner Shipping Timetable Design: A Convex Programming Approach

Container liner shipping is the primary mode of moving manufactured products across continents. Partly due to inherent uncertainties at sea and ports, the liner shipping industry has long had a notorious reputation of schedule delays and unreliable on-time performance. This paper formulates a new approach to incorporate schedule reliability targets in liner shipping timetable design, to balance bunker consumption, time taken for the voyage, and schedule delays. We first model a surrogate problem using a copositive program through a moment decomposition approach and solve it as a convex semidefinite programming relaxation. We next incorporate schedule reliability targets implicitly by exploiting the optimality condition of this surrogate model. We use this approach to analyze the trade-offs between the bunker consumption and the schedule reliability targets for each port call. Furthermore, we derive the optimal speed of the vessel in each leg of the schedule to control for total bunker consumption. Surprisingly, the analytical analysis shows that, if a shipping line has the freedom to choose its preferred berthing times at all ports, it is optimal to set design sailing speeds equal at all legs to maximize schedule reliability. We validate our model using data from a Daily Maersk service, and demonstrate that our schedule can achieve an even higher schedule reliability than the innovative Daily Maersk service schedule, which attained more than 98% reliability in practice, with 11.4% lower bunker consumption. In comparison with a common schedule design heuristic, our model can design service schedules that improve the reliability performance by at least five percentage points, consuming the same amount of bunker. For the same reliability target (of 80%) in schedule design, our model can help reduce bunker consumption by about 11.6% for an 11-week schedule. The savings in bunker consumption can be much more substantial when an ocean carrier aims for higher schedule reliability.

Graph-Constrained Sparse Performance Modeling for Analog Circuit Optimization via SDP Relaxation

In this paper, a graph-constrained sparse performance modeling method is proposed for analog circuit optimization. It builds sparse polynomial models constrained by an acyclic graph. These models can be used to solve analog optimization problems within local design spaces by using convex semidefinite programming relaxation both efficiently and robustly. Our numerical examples demonstrate that the proposed modeling and optimization method can quickly and accurately converge to a superior solution for analog circuits while the conventional method fails to work.

Improved Robust TOA-Based Localization via NLOS Balancing Parameter Estimation

In this paper, the time-of-arrival-based localization problem under mixed line-of-sight/non-line-of-sight (LOS/NLOS) conditions is addressed. Previous studies show that existing robust methods perform well in dense NLOS environments, but generally perform badly in sparse NLOS environments. To alleviate this problem, we introduce a “balancing parameter” related to the NLOS errors and formulate a new robust weighted least squares (RWLS) problem with the source position and the NLOS balancing parameter as the estimation variables. The proposed method does not require the statistics of NLOS errors and the path status. By leveraging the S-Lemma, the RWLS problem is transformed into a non-convex optimization problem, which is then relaxed into a convex semidefinite program. Simulation results show that the proposed method works well for both the sparse and dense NLOS environments.

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.

Cooperative motion parameter estimation using RSS measurements in robotic sensor networks

When multiple mobile targets start from initial positions at constant velocities in a sensor network, a cooperative motion parameter estimation method is proposed by using the received signal strength (RSS) measurements. Firstly, the nonconvex optimization model is built to estimate the initial positions and velocities of the mobile targets. Then the unconstrained semidefinite programming (USDP) is designed to estimate the positions of the mobile targets by converting the nonconvex optimization model into the convex problem, when the transmit powers of the mobile targets are assumed to be known. To reduce the estimation error, the constrained semidefinite programming (CSDP) is also proposed by exploiting the constraint conditions of motion equation. A new convex semidefinite programming (SDP) with known transmit powers (SDP-KTP) is proposed to directly estimate the initial positions and the velocities of the mobile targets. The USDP, CSDP and SDP algorithms are extended to the situation of unknown transmit power. The simulations and real experiments show that the SDP algorithm provides better performance than the USDP or the CSDP by availing of the movement equation. The proposed USDP, CSDP and SDP algorithms perform better with the increasing of samples, but the computational complexity of the proposed algorithms is higher.

Optimal Power Allocation for Mobile Users in Non-Orthogonal Multiple Access Visible Light Communication Networks

In this paper, we focus on the fundamental issues of non-orthogonal multiple access (NOMA) visible light communication (VLC) networks: achievable rates and optimal power allocation schemes for both static and mobile users. First, we derive both a lower bound and an upper bound of the achievable rates with closed-form expressions for static users in NOMA VLC networks. With the derived lower bound, we minimize transmit power under the minimum rate requirements and individual light emitting diodes (LED) power constraints, which turns out to be NP-hard. By exploiting the semidefinite relaxation (SDR) technique, the optimal power allocation scheme can be obtained by solving a convex semidefinite program (SDP). Second, we develop an optimal power allocation scheme for mobile users. Due to users' movement, the estimated channel state information (CSI) may be inaccurate. We first characterize the CSI uncertainties as ellipsoidal regions, and derive a lower bound of the achievable rate expression. Then, we study the transmit power minimization problem for mobile users, which is non-convex. By applying S-lemma and SDR, the transmit power minimization problem can be reformulated as a convex SDP. Simulation results are presented to verify the effectiveness and robustness of the proposed power allocation schemes.

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.

Capacity Bounds and Interference Management for Interference Channel in Visible Light Communication Networks

In this paper, we investigate the channel capacity region of interference channel and develop both centralized and distributed interference management schemes for visible light communication (VLC) networks. For a typical multiuser and multi-LED scenario, we derive both discrete inner and outer bounds of the channel capacity region, and such a proposed inner bound is numerically shown to be the highest among the existing inner bounds. Moreover, with continuous input signals, we develop the channel capacity region bounds in a closed form, termed $\left ({{\alpha,\beta,\gamma } }\right)$ (ABG) inner bound and ABG outer bound, which are tight for the large amplitude-to-variance ratio. Then, based on the derived ABG inner bounds, we investigate a centralized beamforming design problem to minimize the total transmit power under three practical constraints: peak optical power, average optical power, and average electrical power. By utilizing semidefinite relaxation technique, we reformulate this NP-hard problem as a convex semidefinite program and obtain the optimal beamformers. Furthermore, to reduce the cost of channel station information exchange, we propose a distributed coordinated interference management scheme by adopting the alternating direction method of multipliers method. Finally, numerical results are presented to evaluate the performance of the proposed interference management schemes in VLC networks.

Physical-Layer Multicasting Design for Downstream G.fast DSL Transmission

This paper studies the physical-layer multicasting design for downstream G.fast digital subscriber line (DSL) transmission, which corresponds to a multi-user multi-tone (i.e., multi-carrier) scenario. The design goal is to maximize the weighted-sum-group-rate (WSGR) under per-line power constraints. First, as an information-theoretic upper bound, full-rank precoding-based multicasting is considered with joint channel coding across tones. For a single multicast group, this problem corresponds to a non-linear convex semidefinite program (SDP), which is coupled across tones. To reduce the computational complexity, a Lagrange dual decomposition method is developed. This approach is then extended toward multiple multicast groups based on difference-of-convex (DC) programming. Furthermore, a practical multicasting scheme is considered based on rank-one single-stream precoding and independent per-tone channel coding. For this case, instead of relying on computationally complex semidefinite relaxation, a successive convex approximation-based trust-region algorithm is developed. Finally, the simulations of a G.fast cable binder show that the practical multicasting scheme operates close to the information-theoretic multicasting upper bound.

Linear closed-form estimator for sensor localisation using RSS and AOA measurements

Using the hybrid received signal strength (RSS) and angle of arrival (AOA) measurements, a position estimation model is proposed for sensor localisation in three-dimensional plane. Then the unconstraint linear least square (ULLS) and constraint linear least square (CLLS) estimators are designed to obtain the closed-form solutions to the positions of source nodes by considering the known transmit power. When the transmit power is unavailable, a global linear least square (GLLS) estimator is also put forward to estimate the positions of source nodes along with the transmit power. The simulations show that the computational complexity of the proposed linear estimators is greatly lower than that of the convex semidefinite programming (SDP) method. When the measurement noises are small, the linear ULLS, CLLS and GLLS estimators perform better than that of the SDP method. Due to the exploiting of constraint condition, the accuracy performance of the CLLS estimator can approach the Cramer-Rao lower bound (CRLB) of position estimation.

ACOPF for three‐phase four‐conductor distribution systems: semidefinite programming based relaxation with variable reduction and feasible solution recovery

Emerging distribution systems with a proliferation of distributed energy resources are facing with new challenges, such as voltage collapse and power flow congestion in unsymmetrical network configurations. As a fundamental tool that could help quantify these new challenges and further mitigate their impacts on the secure and economic operation of distribution systems, effective AC optimal power flow (ACOPF) models and solution approaches are in urgent need. This study focuses on ACOPF of three-phase four-conductor configured distribution systems, in which neutral conductors and ground resistances are modelled explicitly to reflect practical situation. In addition, by leveraging the Kirchhoff's current law (KCL) theorem and the effect of zero injections, voltage variables of neutrals and zero-injection phases can be effectively eliminated. The ACOPF problem is formulated as a convex semidefinite programming (SDP) relaxation model in complex domain. In recognising possible solution inexactness of SDP relaxation model, a Karush-Kuhn-Tucker condition based process is further proposed to effectively recover feasible solutions to the original ACOPF problem by calculating a set of computational-inexpensive non-linear equations. Numerical studies on a modified IEEE 123-bus system show the effectiveness of the proposed SDP relaxation model with variable reductions and the feasible solution recovery process for three-phase four-conductor configured distribution systems.

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.

A strategy of global convergence for the affine scaling algorithm for convex semidefinite programming

The affine scaling algorithm is one of the earliest interior point methods developed for linear programming. This algorithm is simple and elegant in terms of its geometric interpretation, but it is notoriously difficult to prove its convergence. It often requires additional restrictive conditions such as nondegeneracy, specific initial solutions, and/or small step length to guarantee its global convergence. This situation is made worse when it comes to applying the affine scaling idea to the solution of semidefinite optimization problems or more general convex optimization problems. In (Math Program 83(1–3):393–406, 1998), Muramatsu presented an example of linear semidefinite programming, for which the affine scaling algorithm with either short or long step converges to a non-optimal point. This paper aims at developing a strategy that guarantees the global convergence for the affine scaling algorithm in the context of linearly constrained convex semidefinite optimization in a least restrictive manner. We propose a new rule of step size, which is similar to the Armijo rule, and prove that such an affine scaling algorithm is globally convergent in the sense that each accumulation point of the sequence generated by the algorithm is an optimal solution as long as the optimal solution set is nonempty and bounded. The algorithm is least restrictive in the sense that it allows the problem to be degenerate and it may start from any interior feasible point.

A Physical-layer Security Scheme Based on Cross-layer Cooperation in Dense Heterogeneous Networks

In this paper, we investigate secure communication with the presence of multiple eavesdroppers (Eves) in a two-tier downlink dense heterogeneous network, wherein there is a macrocell base station (MBS) and multiple femtocell base stations (FBSs). Each base station (BS) has multiple users. And Eves attempt to wiretap a macrocell user (MU). To keep Eves ignorant of the confidential message, we propose a physical-layer security scheme based on cross-layer cooperation to exploit interference in the considered network. Under the constraints on the quality of service (QoS) of other legitimate users and transmit power, the secrecy rate of system can be maximized through jointly optimizing the beamforming vectors of MBS and cooperative FBSs. We explore the problem of maximizing secrecy rate in both non-colluding and colluding Eves scenarios, respectively. Firstly, in non-colluding Eves scenario, we approximate the original non-convex problem into a few semi-definite programs (SDPs) by employing the semi-definite relaxation (SDR) technique and conservative convex approximation under perfect channel state information (CSI) case. Furthermore, we extend the frame to imperfect CSI case and use the Lagrangian dual theory to cope with uncertain constraints on CSI. Secondly, in colluding Eves scenario, we transform the original problem into a two-tier optimization problem equivalently. Among them, the outer layer problem is a single variable optimization problem and can be solved by one-dimensional linear search. While the inner-layer optimization problem is transformed into a convex SDP problem with SDR technique and Charnes-Cooper transformation. In the perfect CSI case of both non-colluding and colluding Eves scenarios, we prove that the relaxation of SDR is tight and analyze the complexity of proposed algorithms. Finally, simulation results validate the effectiveness and robustness of proposed 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.