Rank-one Relaxation Research Articles

Weighted Sum-Rate Maximization in Multi-IRS-Aided Multi-Cell mmWave Communication Systems for Suppressing ICI

Intelligent reflecting surface (IRS) has been envisioned as a disruptive technology with the capability of reconfiguring the wireless transmission environment. Hence, the IRS is used to assist the communication system, however, the impacts of inter-cell interference (ICI) on cell-edge users are severe in multi-cell communications but have been ignored in most literature about IRS-assisted communication systems. In this paper, we consider the downlink transmission of cell-edge users in multi-cell millimeter wave (mmWave) communication systems with the aid of multiple IRSs deployed at the cell boundary. To suppress the ICI, we maximize the weighted sum-rate (WSR) by jointly optimizing the IRS-user association matrix, the transmit beamforming of mmWave base stations (MBSs), and the phase shifts of IRSs, subject to the power limits of MBSs, the reflection constraints of IRSs, and the association constraints. Then, a joint IRS-user association and beamforming algorithm with alternating optimization (AO) framework is developed to tackle the mixed-integer non-convex problem. Specifically, a simplified objective function with respect to the association variables is derived and the non-convex binary constraints are handled by the relaxation operation and the first-order Taylor expansion, while the arithmetic-geometric mean inequality (AGMI)-based rank-one relaxation algorithm and the ring-based penalty convex-concave procedure (CCP) method are proposed for obtaining the transmit beamforming matrix of MBSs and the phase shifts of IRSs, respectively. Simulation results demonstrate the superiority of the proposed algorithm over the benchmark schemes. The useful insights into the optimization of IRS-user association and design of the number of elements of IRSs are also presented for future networks.

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, our experiments showed that the relaxation by rank-one convexification delivering an approximation to the quasiconvex envelope prevents mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

Mathbf {2times 2}-Convexifications for convex quadratic optimization with indicator variables

In this paper, we study the convex quadratic optimization problem with indicator variables. For the {2times 2} case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the {2times 2} case as a building block, we derive an extended SDP relaxation for the general case. This new formulation is stronger than other SDP relaxations proposed in the literature for the problem, including the optimal perspective relaxation and the optimal rank-one relaxation. Computational experiments indicate that the proposed formulations are quite effective in reducing the integrality gap of the optimization problems.

Transceiver coexistence design of MIMO radar and MIMO communication under Gaussian model uncertainty

This paper proposes a transceiver coexistence design of multiple-input multiple-output (MIMO) radar and MIMO communication under Gaussian model uncertainty. To tackle the mutual interference and the model uncertainty problems, the transceivers of the radar system and the communication system are jointly designed to minimise the total transmit power of the radar system and the communication system, while the signal-to-interference-plus-noise-ratios (SINRs) or effective SINR (ESINR) of the radar system and the mean square error (MSE) of the communication system are guaranteed with small outage probability. Safe approximations are proposed for the probabilistic constraints, and a converged iterative algorithm is proposed for the joint transceiver design. Owing to the MSE formulation, the convex relaxation for the communication transmitter is proved to be tight. Furthermore, the tight communication relaxation provides an effective way to check whether the radar error is majorly due to numeric error or the possible rank-one relaxation error. Extensive simulation results show that the proposed iterative algorithm converges, the possible radar rank-one relaxation error is trivial with respect to the numeric error, and the quality of service in terms of SINR, ESINR, MSE and bit error rate (BER) in the proposed coexistence system are robust against model uncertainty with the cost of using extra transmit power.

Robust Secure Beamforming Design for Cooperative Cognitive Radio Nonorthogonal Multiple Access Networks

By the integration of cooperative cognitive radio (CR) and nonorthogonal multiple access (NOMA), cooperative CR NOMA networks can improve the spectrum efficiency of wireless networks significantly. Due to the openness and exposure of wireless signals, secure communication is an important issue for cooperative CR NOMA networks. In this paper, we investigate the physical layer security design for cooperative CR NOMA networks. Our objective is to achieve maximum secrecy rate of the secondary user by designing optimal beamformers and artificial noise covariance matrix at the multiantenna secondary transmitter under the quality-of-service at the primary user and the transmit power constraint at the secondary transmitter. We consider the practical case that the channel state information (CSI) of the eavesdropper is imperfect, and we model the imperfect CSI by the worst-case model. We show that the robust secrecy rate maximization problem can be transformed to a series of semidefinite programmings based on S-procedure and rank-one relaxation. We also propose an effective method to recover the optimal rank-one solution. Simulations are provided to show the effectiveness of our proposed robust secure algorithm with comparison to the nonrobust secure design and traditional orthogonal multiple access schemes.

Optimized Two-Time Scale Robust Dispatching Method for the Multi-Terminal Soft Open Point in Unbalanced Active Distribution Networks

To alleviate the impact of output uncertainty of renewable energy sources (RESs) in unbalanced active distribution networks (ADNs), this paper proposed a two-time scale robust optimization method for the multi-terminal soft open point (SOP). In the long-term scale, the operation points of the SOP were optimized by semidefinite programming (SDP) model to minimize system loss and mitigate voltage unbalance. An improved iterative cutting plane (ICP) method was proposed to strengthen the exactness of the SDP rank-one relaxation. In the short-term scale, V 2- P and V 2- Q droop control mode was implemented in the SOP to better respond to the output uncertainty of RESs. Then a slope robust optimization model was established based on long-term operation points and short-term forecast data to improve system robust security. To coordinate these two time scales, a co-optimization re-dispatch strategy was proposed in the short-term model so that the robust safety margin could be further expanded in extreme conditions. Case studies show that the proposed method can fully utilize the flexible power flow regulation ability of the multi-terminal SOP to reduce system loss, mitigate voltage unbalance and guarantee system robust security in the presence of RES output uncertainty.

Full-duplex wireless-powered jammer aided secure communication for cognitive radio networks

A novel full-duplex (FD) wireless-powered cooperative jammer (CJ) aided transmit scheme is proposed to enhance the security for cognitive radio networks (CRNs). The CJ operates in an FD mode, so that it can harvest energy from the signals that transmitted from the secondary transmitter (ST), and send jamming noise (JN) to confuse the potential malicious eavesdroppers in the meantime. We aim to maximize the achievable secrecy rate of the secondary system by jointly designing the transmit covariance matrix at the ST and the JN covariance matrix at the CJ, while subject to the transmit power constraints at the ST and the CJ, and the interference constraint at the primary user (PU). Specifically, we formulate the secrecy rate maximization problem based on the perfect channel state information (CSI). As a solution, we propose a two-stage optimization algorithm to solve the non-convex problem. More particularly, we extend our proposed scheme to the imperfect CSI case. To tackle the robust secrecy rate maximization problem, we exploit the two-stage optimization algorithm with the extended S-procedure and rank-one relaxation. Simulation results have validated that our proposed scheme outperforms, in terms of secrecy rate, the isotropic JN scheme and the scheme without aided by CJ for the perfect CSI scenario. In addition, the robust scheme achieves higher secrecy rate gains as compared to the non-robust scheme for the imperfect CSI case.

Secrecy Rate Optimization for Cooperative Cognitive Radio Networks Aided by a Wireless Energy Harvesting Jammer

In this paper, a novel wireless energy harvesting cooperative jammer (EH-CJ)-aided transmission scheme, which can enhance the security for cooperative cognitive radio networks (CCRNs), is proposed. The EH-CJ first harvests energy from ambient radio frequency (RF) signals transmitted from the primary transmitter (PT). This energy is then used to transmit jamming noise (JN), so that the performance of eavesdroppers’ receptions is degraded, and also to deliver additional power to the secondary energy receiver (ER). In this way, the proposed scheme can enhance the security and extend the lifetime for CCRNs. For this system, we formulate an optimization problem for maximizing the secrecy rate of the secondary system under transmit power constraints at the secondary transmitter as well as the EH-CJ, minimum harvested energy at the secondary ER and information rate at the primary receiver (PR). Since this optimization problem is not convex, we propose a two-stage optimization algorithm based upon the Charnes–Cooper transformation and rank-one relaxation to solve it. Computer simulation results have demonstrated that the proposed EH-CJ-aided scheme achieves better secrecy rate performance, as compared to an isotropic JN scheme as well as compared to an equivalent scheme which does not use the aid of EH-CJ.

Robust Beamforming for Simultaneous Wireless Information and Power Transfer in MISO Interference Channels

Taking the channel uncertainty into account, we investigate the robust beamforming design for simultaneous wireless information and power transfer in multiple-input single-output interference channels, where multiple multi-antenna transmitters communicate with their targeted single-antenna receivers and simultaneously interfere with all other receivers. Under quality-of-service and energy harvesting constraints at the receivers which adopt power splitting scheme, we aim to minimize the total transmit power at the transmitters when the channel uncertainty is modeled as the norm-bounded. The robust optimization is non-convex and the global optimal solution is challenging. With the help of the extension of the S-Procedure and rank-one relaxation, the robust beamforming is transformed into a semidefinite programming . We develop an novel method to recover a rank-one feasible and locally optimal solution, which is based on difference of convex programming and sequential parametric convex approximation method. Simulations show that our proposed algorithm outperforms the well-known Gaussian randomization method and has much lower computational complexity.

Beamforming for Information and Energy Cooperation in Cognitive Non-Regenerative Two-Way Relay Networks

In this paper, we investigate information and energy cooperation in cognitive non-regenerative two-way relay networks, where the multi-antenna secondary user (SU) transmitter harvests energy from primary users’ (PUs) signals by the power splitting (PS) scheme to forward PUs’ signals and transmit its own signals. Our objective is to design beamforming matrix and vector at the SU transmitter and PS factor to maximize achievable rate at SUs while maintaining achievable rate requirements at PUs subject to transmit power constraint at SU transmitter. We consider two scenarios that the SU transmitter knows perfect and imperfect channel state information of all links. For the former, we propose an optimal solution based on Charnes-Cooper transformation and 1-D search. We also propose a low-complexity suboptimal solution based on the zero-forcing (ZF) scheme and the algebraic norm-maximizing scheme. For the latter, channel uncertainties are modeled by a worst-case model. We propose to employ a ZF scheme and transform the problem into a semidefinite programming with the help of $\mathcal {S}$ -procedure and rank-one relaxation. The rank-one feasible solution is proposed to be recovered by the Gaussian randomization method. Simulation results demonstrate that our proposed designs outperform the beamforming scheme without energy cooperation.

Robust Design of Transmit Waveform and Receive Filter For Colocated MIMO Radar

We consider the problem of angle-robust joint transmit waveform and receive filter design for colocated multiple-input multiple-output (MIMO) radar, in the presence of signal-dependent interferences. The design problem is cast as a max-min optimization problem to maximize the worst-case output signal-to-interference-plus-noise-ratio (SINR) with respect to the unknown angle of the target of interest. Based on rank-one relaxation and semi-definite programming (SDP) representation of a nonnegative trigonometric polynomial, a cyclic optimization algorithm is proposed to tackle this problem. The effectiveness of the proposed method is illustrated via numerical examples.

Numerical energy relaxation to model microstructure evolution in functional magnetic materials

AbstractThis paper proposes energy relaxation‐based approaches for the modeling of magnetostriction, with a particular focus on single crystalline magnetic shape memory alloy response. The theoretical development relies on concepts of energy relaxation in the context of nonconvex free energy landscapes whose wells define preferred states of spontaneous straining and magnetization. The constrained theory of magnetoelasticity developed by DeSimone and James [1] represents the point of departure for the model development, and its capabilities, but also limitations, are demonstrated by means of representative numerical examples. The key features that characterize the extended approach are (i) the incorporation of elastic deformations, whose distribution among the individual phases occurs in an energy minimizing fashion, (ii) a finite magnetocrystalline anisotropy energy, that allows magnetization rotations away from easy axes, and (iii) dissipative effects, that are accounted for in an incremental variational setting for standard dissipative materials. In the context of introducing elastic strain energy, two different relaxation concepts, the convexification approach and the rank‐one relaxation with respect to first‐order laminates, are considered. In this manner, important additional response features, e.g. the hysteretic nature, the linear magnetization response in the pre‐variant reorientation regime, and the stress dependence of the maximum field induced strain, can be captured, which are prohibited by the inherent assumptions of the constrained theory. The enhanced modeling capabilities of the extended approach are demonstrated by several representative response simulations and comparison to experimental results taken from literature. These examples particularly focus on the response of single crystals under cyclic magnetic field loading at constant stress and cyclic mechanical loading at constant magnetic field. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A novel homogenization method for phase field approaches based on partial rank-one relaxation

This paper deals with the analysis of homogenization assumptions within phase field theories in a finite strain setting. Such homogenization assumptions define the average bulk׳s energy within the diffusive interface region where more than one phase co-exist. From a physical point of view, a correct computation of these energies is essential, since they define the driving force of material interfaces between different phases. The three homogenization assumptions considered in this paper are: (a) Voigt/Taylor model, (b) Reuss/Sachs model, and (c) Khachaturyan model. It is shown that these assumptions indeed share some similarities and sometimes lead to the same results. However, they are not equivalent. Only two of them allow the computation of the individual energies of the co-existing phases even within the aforementioned diffusive interface region: the Voigt/Taylor and the Reuss/Sachs model. Such a localization of the averaged energy is important in order to determine and to subsequently interpret the driving force at the interface. Since the Voigt/Taylor and the Reuss/Sachs model are known to be relatively restrictive in terms of kinematics (Voigt/Taylor) and linear momentum (Reuss/Sachs), a novel homogenization approach is advocated. Within a variational setting based on (incremental) energy minimization, the results predicted by the novel approach are bounded by those corresponding to the Voigt/Taylor and the Reuss/Sachs model. The new approach fulfills equilibrium at material interfaces (continuity of the stress vector) and it is kinematically compatible. In sharp contrast to existing approaches, it naturally defines the mismatch energy at incoherent material interfaces. From a mathematical point of view, it can be interpreted as a partial rank-one convexification.

A Special Class of Fractional QCQP and Its Applications on Cognitive Collaborative Beamforming

In this paper, we investigate a special class of the fractional quadratically constrained quadratic problem (QCQP) with more than two quadratic constraints. We propose to effectively solve this special fractional QCQP by one or two convex semidefinite programmings (SDPs). For the two SDPs, one is equivalent to the original fractional QCQP with rank-one relaxation from the Charnes-Cooper transformation and the other, exploiting the optimal value of the former SDP, always has rank-one solution, which is optimal to the former SDP. Theoretical analysis shows that our proposed non-iterative SDP-based algorithm achieves the global optimal solution to the special fractional QCQP. In specific scenarios, our proposed non-iterative SDP-based algorithm has lower computational complexity compared to the second-order cone programming (SOCP)-based and constrained concave convex procedure (CCCP)-based iterative algorithms. We apply the proposed non-iterative SDP-based algorithm on two collaborative beamforming problems in cognitive relay networks, specifically, one is the achievable rate region for two-way non-regenerative cognitive relay networks and the other is the achievable secrecy rate for one-way regenerative cognitive relay networks. Simulation results have shown that our proposed non-iterative SDP-based algorithm achieves the same performance as the SOCP-based iterative algorithm. Our proposed algorithm achieves the better performance than the CCCP-based iterative algorithm.

Relay Beamforming for Amplify-and-Forward Multi-Antenna Relay Networks with Energy Harvesting Constraint

For amplify-and-forward multi-antenna relay networks with energy harvesting (EH) constraint, we study the optimal relay beamforming problem which maximizes the achievable rate from source to information-decoding receiver subject to the transmit power constraint at relay and the EH constraint at EH receiver. Because of the EH constraint, the beamforming problem is not convex. We propose the optimal beamforming scheme by converting the beamforming problem into a convex semidefinite programming with the rank-one relaxation and Charnes-Cooper transformation. We also propose a suboptimal closed-form beamforming scheme. It is shown from simulations that when the maximum allowable relay transmit power to noise power ratio is high, the performance of proposed suboptimal scheme approaches that of the optimal scheme.

Robust Beamforming for Cognitive Multi-Antenna Relay Networks with Bounded Channel Uncertainties

In cognitive relay networks, the interferences from secondary users (SUs) and relays to primary users are constrained to be lower than a threshold. The interference constraints are difficult to satisfy when the channel state information (CSI) is imperfect. In this paper, we propose a robust beamforming scheme for the multi-antenna non-regenerative cognitive relay network where the multi-antenna relay with imperfect CSIs helps the communication of single-antenna SU. Our objective is to design a robust beamforming scheme which maximizes the system capacity subject to transmit power constraint and interference constraints. The bounded channel uncertainties are modeled using the worst-case model. The robust beamforming problem, neglecting the correlation of channel uncertainties, is reformulated as a convex semidefinite programming (SDP) by rank-one relaxation. This convex SDP is related with the worst-case relay transmit power minimization problem, which is further reformulated as a convex SDP, whose rank-one solution is proved to exist. Thus, we propose the suboptimal solution to the robust beamforming problem which is found effectively by solving two convex SDPs. Simulation results are provided to demonstrate the effectiveness of the proposed scheme.

Optimal beamforming in spectrum sharing cognitive non‐regenerative multi‐relay networks

An optimal beamforming design for spectrum sharing cognitive non-regenerative multi-relay networks that maximises the signal-to-noise ratio of the secondary user subject to the total transmit power constraint at the relays and the interference power constraint at the primary user is proposed. In the power-limited regime and the interference-limited regime, closed-form beamformers can be found. While in the intermediate operating regime, the beamforming problem is first transformed to a convex semidefinite programming (SDP) via rank-one relaxation and Charnes-Cooper transformation, and then the optimal beamformers are found by applying the rank-one decomposition theorem to the solution of the established SDP. Simulation results show that the proposed solution performs better than existing schemes.

Optimal Relay Selection and Beamforming in MIMO Cognitive Multi-Relay Networks

For the non-regenerative multiple-input multiple-output (MIMO) cognitive multi-relay network, we propose an optimal relay selection and beamforming scheme which maximizes the capacity of the secondary user by selecting the best cognitive MIMO relay subject to the transmit power constraints at the relays and the interference power constraints at the primary users. In our proposed scheme, the relay selection and beamforming optimization problem is solved separately by employing convex semidefinite programming (SDP) through rank-one relaxation and Charnes-Cooper transformation. It is shown that our proposed scheme achieves higher average capacity than the convectional relay beamforming schemes and relay selection schemes.

New semidefinite programming relaxations for box constrained quadratic program

We establish in this paper optimal parametric Lagrangian dual models for box constrained quadratic program based on the generalized D.C. (difference between convex) optimization approach, which can be reformulated as semidefinite programming problems. As an application, we propose new valid linear constraints for rank-one relaxation.