Solution Of Semidefinite Relaxation Research Articles

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.

Localization Using Time-Delay and Doppler Shift by Moving Monostatic Sensors

This article considers the localization of a stationary target using time-delay and Doppler shift measurements obtained by moving monostatic sensors that takesthe sensor motion during the localization period into consideration. Starting from the model equation of a previous work, we propose a new formulation of optimization through constrained weighted least-squares to solve for the target location by developing two new localization solutions, one in closed form and the other by semi-definite relaxation (SDR). The new formulation enables the reduction of the minimum number of auxiliary variables in the closed-form solution and allows additional constraints in the SDR solution, resulting in fewer sensors needed in the former and less relaxation error and higher noise tolerance in the latter. Simulations validate the better performance of the proposed solutions than the previous ones.

Reconfigurable Intelligent Surface-Assisted Non-Orthogonal Multiple Access

Reconfigurable intelligent surface (RIS) is a revolutionary technology to achieve spectrum-, energy-, and cost-efficient wireless networks. This paper considers an RIS-assisted downlink non-orthogonal-multiple-access (NOMA) system. To optimize the rate performance and ensure user fairness, we maximize the minimum decoding signal-to-interference-plus-noise-ratio (equivalently the rate) of all users, by jointly optimizing the (active) transmit beamforming at the base station (BS) and the phase shifts (i.e., passive beamforming) at the RIS. A combined-channel-strength based user-ordering scheme for NOMA decoding is first proposed to decouple the user-ordering design and the joint beamforming design. Efficient algorithms are further proposed to solve the non-convex problem, by leveraging the block coordinated descent and semidefinite relaxation (SDR) techniques. For the single-antenna BS setup, the optimal power allocation at the BS and the asymptotically optimal phase shifts at the RIS are obtained in closed forms. For the multiple-antenna BS setup, it is shown that the rank of the SDR solution of the transmit beamforming design is upper bounded by two. Also, the proposed algorithms are analyzed in terms of convergence and complexity. Simulation results show that the RIS-assisted NOMA system can enhance the rate performance significantly, compared to traditional NOMA without RIS and traditional orthogonal multiple access with/without RIS.

Distributionally Robust Secure Multicast Beamforming With Intelligent Reflecting Surface

Recently, the intelligent reflecting surfaces (IRSs)-aided wireless transmission has drawn considerable attention. This paper investigates the use of IRS in enhancing the physical-layer security of the multiuser multiple-input single-output (MU-MISO) broadcast system, where a base station (BS) transmits a common data stream to multiple legitimate receivers in the presence of multiple eavesdroppers (Eves). The BS is assumed to have only some erroneous channel state information (CSI) of the Eves. The CSI error is modeled by a moment-based random error model, in which the BS only knows the first- and second-order statistics of the error, but not the exact distribution. Under this CSI error model, we investigate the joint robust design of the secure beamforming at the BS and the phase shift at the IRS to maximize the worst legitimate user’s SNR, while keeping the Eves’ SNR below certain threshold with high probability, evaluated with respect to any distribution fulfilling the given first and second-order statistics. This robust secure transmission design problem is a semi-infinite chance-constrained problem, which is in general intractable. We show that the considered problem admits an equivalent conic reformulation, which can be handled by alternately optimizing the beamformer at the BS and the phase shift at the IRS via semidefinite relaxation (SDR) and penalty convex-concave procedure (CCP), respectively. To further improve the secrecy performance, we also study another rank-two beamformed Alamouti transmission scheme at the BS, which can be seen as a generalization of conventional rank-one beamforming. Simulation results demonstrate that the proposed designs are robust against the error distribution, and that the inclusion of IRS is not only helpful for enhancing the security, but also useful for promoting a low-rank SDR solution.

Joint transmitter and PS factor optimal design for secure transmission in MIMO full‐duplex relay network powered by SWIPT with perfect and imperfect channel state information

The joint transmitter and PS factor optimal design scheme for secure transmission in multiple-input multiple-output (MIMO) full-duplex relay network powered by simultaneous wireless information and power transfer (SWIPT) is studied. An optimisation problem is presented, aimed at maximising the system security capacity by jointly optimising transmitter and PS factor while satisfying the energy harvest constraint in both perfect and imperfect channel state information (CSI). In perfect CSI case, the optimisation function is formulated with energy harvest constraint, and semidefinite relaxation (SDR) is introduced to solve the non-convex optimisation problem. In imperfect CSI case, it is assumed that only the first and second order statistics of the eavesdropper's channels are available, but the specific distribution is unknown. In this case, the security capacity of the system is maximised while satisfying the security outage probability and energy harvest constraint. The SDR solution based on equivalent conic reformulation is proposed. And the simulation results demonstrate the effectiveness of the optimal designs in both CSI cases.

Effect of Sensor Motion on Time Delay and Doppler Shift Localization: Analysis and Solution

This paper investigates active localization of a stationary object using time delay alone or together with Doppler shift measurements observed by a number of dynamic monostatic sensors having collocated transmitters and receivers, where the sensor motion during the interrogation period is not negligible. Non-negligible motion effect appears when the sensor speed relative to the signal propagation speed is significant, such as in an acoustic or an underwater environment. The motion effect causes the signal sent and received locations different, with their separation proportional to the two-way propagation time delay that is dependent on the object location. We first provide new measurement models for time delay and Doppler shift in the presence of sensor motion. They come as recursive equations and their non-recursive forms are derived. Analysis of the motion effect based on the Cramer-Rao lower bound (CRLB) is conducted, and the performance loss in terms of the location bias and variance when ignoring the sensor motion effect is evaluated. Two kinds of estimators for the object location are next proposed by exploiting the non-recursive forms of the model equations. One is a closed-form solution by algebraic evaluation and the other is a semi-definite relaxation solution by convex optimization. The first is computationally efficient and is shown to achieve the CRLB performance over the small error region under Gaussian noise, while the second can handle the positioning problem better when the noise level is high. Simulations validate the theoretical analysis and the performance of the proposed estimators.

Full-Duplex Secure Relay Beamforming Design for Systems With Perfect and Partial CSI

Considering that wireless unidirectional full-duplex (FD) transmission is one of the FD transmissions over 5G mobile wireless networks, this paper proposes a joint receive and transmit relay beamformer design for systems consisting of a source, a FD relay, an intended destination, and an eavesdropper. The source and destination are single-antenna nodes, whereas the FD relay is a multi-antenna node. Assuming that full channel state information (CSI) is available and the eavesdropper processes two blocks of received data at a time, the joint optimization problem is reformulated as a semidefinite relaxation (SDR) problem (in terms of relay transmit beamformer), which is tight in the sense that the rank-one beamformer can always be recovered from the SDR solution. In order to significantly minimize the computational cost of the SDR implementation, a novel semi-analytical approach based on dual optimization is proposed to obtain the optimum transmit beamformer. When the number of data blocks is arbitrary, we propose a successive convex approximation approach for solving the original non-convex optimization problem and show that a stationary solution can be efficiently obtained. We then propose a distributionally robust beamformer design for the case of an arbitrary number of blocks, assuming that the means and covariance matrices of the eavesdropper's channels are known. Computer simulations show that the proposed semi-analytical solution matches with that obtained from numerical optimization of the SDR problem, the FD scheme significantly outperforms its half-duplex counterpart, and the proposed robust design provides robustness against CSI uncertainty.

Parallel Jobs Scheduling with a Specific Due Date: Asemi-definite Relaxation-based Algorithm

This paper considers a different version of the parallel machines scheduling problem in which the parallel jobs simultaneously requirea pre-specifiedjob-dependent number of machines when being processed.This relaxation departs from one of the classic scheduling assumptions. While the analytical conditions can be easily statedfor some simple models, a graph model approach is required when conflicts of processor usage are present. The main decisions and solving steps are as follows, respectively. (i) Converting the scheduling problem to graph model; (ii) Dividing jobs into independent sets: in this phase, we propose a semi-definite relaxation algorithm in which we use graph coloring concept; (iii) Sequencing the independent sets as a single-machine scheduling in which jobs in such a system arejob sets formed by using a semi-definite relaxation solution and determining the problem as a schedule that minimizes the sum of the tardiness of jobs. In this regard, after grouping the jobs by a semi-definite programming relaxation algorithm, we used the rounding algorithm for graph coloring. We also proposed a variable neighborhood search algorithm for sequencing the obtained job sets in order to minimize the sum of the tardiness. Experimental results show that this methodology is interesting by obtaining good results.

Robust secrecy beamforming for full-duplex two-way relay networks under imperfect channel state information

Consider a two-way amplify-and-forward (AF) relay network where two legitimate nodes communicate through a relay in the presence of an eavesdropper. Assuming full duplexity at the legitimate nodes and the relay, a robust artificial noise (AN)-aided AF scheme is proposed to maximize the worst-case sum secrecy rate under imperfect channel state information (CSI) of the eavesdropper. This robust sum secrecy rate maximization (SSRM) problem is formulated as a max-min semi-infinite problem and is tackled by the semidefinite relaxation (SDR) method. In particular, we first convert the max-min semi-infinite problem into a maximization problem with a finite number of constraints. Then, an efficient two-block alternating difference-of-concave (DC) programming approach is proposed to iteratively solve the SDR problem, with one of the blocks computed in closed form. In addition, a specific robust rank-one solution construction procedure is presented to extract a feasible solution for the original robust SSRM problem from the SDR solution. The efficacy of the proposed method is demonstrated by numerical simulations.

General Rank Multiuser Downlink Beamforming With Shaping Constraints Using Real-Valued OSTBC

In this paper we consider optimal multiuser downlink beamforming in the presence of a massive number of arbitrary quadratic shaping constraints. We combine beamforming with full-rate high dimensional real-valued orthogonal space time block coding (OSTBC) to increase the number of beamforming weight vectors and associated degrees of freedom in the beamformer design. The original multi-constraint beamforming problem is converted into a convex optimization problem using semidefinite relaxation (SDR) which can be solved efficiently. In contrast to conventional (rank-one) beamforming approaches in which an optimal beamforming solution can be obtained only when the SDR solution (after rank reduction) exhibits the rank-one property, in our approach optimality is guaranteed when a rank of eight is not exceeded. We show that our approach can incorporate up to 79 additional shaping constraints for which an optimal beamforming solution is guaranteed as compared to a maximum of two additional constraints that bound the conventional rank-one downlink beamforming designs. Simulation results demonstrate the flexibility of our proposed beamformer design.

Approximation Bounds for Semidefinite Relaxation of Max-Min-Fair Multicast Transmit Beamforming Problem

Consider a downlink multicast scenario where a base station equipped with multiple antennas wishes to simultaneously broadcast a number of signals to some given groups of users over a common bandwidth. The goal of the base station is to select appropriate beamforming vectors so as to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among all users under a power budget constraint. Since this max-min-fair transmit beamforming problem is NP-hard in general, a randomized polynomial time approximation approach based on semidefinite relaxation (SDR) has been proposed recently where excellent performance in both simulated and measured wireless channels has been reported. This paper shows that the SDR-based approach can provide at least an O(1/M) approximation to the optimum solution, where M is the total number of users. This estimate implies that the SDR solution achieves an SINR that is at most (logM+O(1)) dB away from the highest possible value. The existence of such a data independent bound certifies the worst-case approximation quality of the SDR algorithm for any problem instance and any number of transmit antennas. For real-valued problems, the corresponding approximation ratio is shown to be O(1/M2), while the SINR loss due to SDR approximation is at most (2logM+O(1)) dB.

On the gap between the quadratic integer programming problem and its semidefinite relaxation

Consider the semidefinite relaxation (SDR) of the quadratic integer program (QIP): ** where Q is a given symmetric matrix and D is diagonal. We consider the SDR gap **. We establish the uniqueness of the SDR solution and prove that ** if and only if γr:=n−1max{xTVVTx:x ∈ {-1, 1}n}=1 where V is an orthogonal matrix whose columns span the (r–dimensional) null space of D−Q and where D is the unique SDR solution. We also give a test for establishing whether ** that involves 2r−1 function evaluations. In the case that γr<1 we derive an upper bound on γ which is tighter than **. Thus we show that `breaching' the SDR gap for the QIP problem is as difficult as the solution of a QIP with the rank of the cost function matrix equal to the dimension of the null space of D−Q. This reduced rank QIP problem has been recently shown to be solvable in polynomial time for fixed r.