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In this letter, we investigate the robust beamforming design for a hybrid active-passive reconfigurable intelligent surface (RIS) assisted integrated sensing and communication (ISAC) system, where the RIS consists of both active and passive reflecting elements. Different from the existing works, we consider the target location estimation error. We aim to maximize the minimum sensing beampattern gain among multiple targets, subject to the signal-to-interference-plus-noise ratio (SINR) constraint for each communication user (CU), the transmit power constraints for the BS and the active elements. To address the non-convexity of the max-min RIS-assisted ISAC design problem, we propose an alternating-optimization based design solution to alternately optimize the BS’s transmit beamforming design and the hybrid RIS’s coefficient matrix in a low-complexity manner. Simulation results demonstrate the benefit of the proposed design scheme over the existing baseline schemes. IEEE

Biological applications of nanomaterials as delivery carriers have been embedded in traditional biomedical research for decades. Despite lagging behind, recent significant breakthroughs in the use of nanocarriers as tools for plant biotechnology have created great interest. In this Perspective, we review the outstanding recent works in nanocarrier-mediated plant transformation and its agricultural applications. We analyze the chemical and physical properties of nanocarriers determining their uptake efficiency and transport throughout the plant body.

This paper considers a $K$-user multiple-input single-output (MISO) interference channels for simultaneous wireless information and power transfer (SWIPT), where each multi-antenna transmitter serves a single-antenna receiver per user pair. All receivers perform simultaneously information processing and energy harvesting (EH) based on the receive power-splitting (PS) architectures. Assuming imperfect channel state information (CSI) at the transmitters, we develop an optimal robust transceiver design scheme that minimizes the total transmission power under the worst-case signal-to-interference-plus-noise ratio (SINR) and energy harvesting (EH) constraints at the receivers, by jointly optimizing transmit beamforming and receive PS ratio per receiver. When the CSI uncertainties are bounded by ellipsoidal regions, it is shown that the worst-case SINR and EH constraints per receiver can be recast into quadratic matrix inequality forms. Leveraging semidefinite relaxation technique, the intended robust beamforming and PS (BFPS) problem can be relaxed as a tractable (centralized) semidefinite program (SDP). More importantly, relying on the state-of-the-art alternating direction method of multipliers (ADMM) in convex optimization, we propose a {\em decentralized} algorithm capable of computing the optimal robust BFPS scheme with local CSI and limited information exchange among the transmitters. It is shown the proposed decentralized algorithm is guaranteed to converge to the optimal centralized solution. Numerical results are provided to demonstrate the merits of the proposed approaches., Comment: 8 pages and 7 figures

The design of a set of beamformers for the multiuser multiple-input single-output (MISO) downlink that provides the receivers with prespecified levels of quality-of-service (QoS) can be quite challenging when the channel state information is not perfectly known at the base station. The constraint of having the SINR meet or exceed a given threshold with high probability is intractable in general, which results in problems that are fundamentally hard to solve. In this paper, we will develop a high quality approximation of the SINR outage constraint that, along with a semidefinite relaxation, enables us to formulate the beamformer design problem as a convex optimization problem that can be efficiently solved. For systems in which the uncertainty size is small, a further approximation yields algorithms based on iterative evaluations of closed-form expressions that have substantially lower computational cost. Since finding the beamforming directions incurs most of the computational load of these algorithms, analogous power loading algorithms for predefined beamforming directions are developed and their performance is shown to be close to optimal. When the system contains a large number of antennas, the proposed power loading can be obtained at a computational cost that grows only linearly in the number of antennas. The proposed power loading algorithm provides an explicit relationship between the outage probability required and the power consumed, which allows us to precisely control the power consumption, and automatically identifies users who are consuming most of the power resources. The flexibility of the proposed approach is illustrated by developing a power loading technique that minimizes an average notion of outage.

Introduction In the spectrum sharing cognitive radio (CR) networks, downlink beamforming design for the secondary transmission has been an intensive research topic in the past decade. Spectrum sharing allows secondary and primary users to access the same channel concurrently, and beamforming techniques can be applied in order to avoid excessive interference caused to the primary users while steering power towards the secondary receivers, and by equipping the secondary transmitter (e.g., a base station or access point) with antenna arrays. For a comprehensive coverage of the recent advances on CR communications and networking, readers are referred to the survey paper [1] and the magazine paper [2]. In particular, [3]– [5] provided readers some recent specific works on optimal CR transmit beamforming.

Introduction In the spectrum sharing cognitive radio (CR) networks, downlink beamforming design for the secondary transmission has been an intensive research topic in the past decade. Spectrum sharing allows secondary and primary users to access the same channel concurrently, and beamforming techniques can be applied in order to avoid excessive interference caused to the primary users while steering power towards the secondary receivers, and by equipping the secondary transmitter (e.g., a base station or access point) with antenna arrays. For a comprehensive coverage of the recent advances on CR communications and networking, readers are referred to the survey paper [1] and the magazine paper [2]. In particular, [3]– [5] provided readers some recent specific works on optimal CR transmit beamforming.

This paper considers the beamforming design for a multiuser multiple-input single-output (MISO) downlink with an arbitrary number of (context-specific) shaping constraints. In this setup, the state-of-the-art beamforming schemes cannot attain the well-known performance bound promised by the semidefinite program (SDP) relaxation technique. To close the gap, we propose a redundant-signal embedded linear beamforming (REEL-BF) scheme, where each user is assigned with one information beamformer and several shaping beamformers. It is shown that the proposed REEL-BF scheme can perform general rank-$K$ beamforming for user symbols in a low-complexity and structured manner. In addition, sufficient conditions are derived to guarantee that the REEL-BF scheme always achieves the SDP bound for linear beamforming schemes. Based on such conditions, an efficient algorithm is then developed to obtain the optimal REEL-BF solution in polynomial time. Numerical results demonstrate that the proposed scheme enjoys substantial performance gains over the existing alternatives., Comment: Accepted for publication in the IEEE Transactions on Signal Processing

We deal with the robust design of the Multiple-Input Multiple-Output (MIMO) waveform covariance matrix optimizing the worst case (over steering mismatches) transmit beampattern Peak Sidelobe Level (PSL). To this end, we model the uncertainty set associated with each steering vector through two double-sided, potentially non-convex, quadratic constraints. Additionally, we force two suitable constraints on the optimization variable. The former accounts for the width of the mainbeam, whereas the latter is either a uniform or a relaxed elemental power requirement allowing to control the amount of transmitted power. We prove that the mentioned design leads to a non-convex optimization problem which remarkably shares some hidden convexity. Hence, we devise a polynomial time procedure aimed at synthesizing an optimal MIMO waveform covariance. Finally, at the analysis stage, we assess the performance of the proposed design technique showing that it ensures an improved worst case performance than some counterparts available in open literature. © 2015 IEEE.

We deal with the robust design of the Multiple-Input Multiple-Output (MIMO) waveform covariance matrix optimizing the worst case (over steering mismatches) transmit beampattern Peak Sidelobe Level (PSL). To this end, we model the uncertainty set associated with each steering vector through two double-sided, potentially non-convex, quadratic constraints. Additionally, we force two suitable constraints on the optimization variable. The former accounts for the width of the mainbeam, whereas the latter is either a uniform or a relaxed elemental power requirement allowing to control the amount of transmitted power. We prove that the mentioned design leads to a non-convex optimization problem which remarkably shares some hidden convexity. Hence, we devise a polynomial time procedure aimed at synthesizing an optimal MIMO waveform covariance. Finally, at the analysis stage, we assess the performance of the proposed design technique showing that it ensures an improved worst case performance than some counterparts available in open literature. © 2015 IEEE.

We deal with the robust design of the Multiple-Input Multiple-Output (MIMO) waveform covariance matrix optimizing the worst case (over steering mismatches) transmit beampattern Peak Sidelobe Level (PSL). To this end, we model the uncertainty set associated with each steering vector through two double-sided, potentially non-convex, quadratic constraints. Additionally, we force two suitable constraints on the optimization variable. The former accounts for the width of the mainbeam, whereas the latter is either a uniform or a relaxed elemental power requirement allowing to control the amount of transmitted power. We prove that the mentioned design leads to a non-convex optimization problem which remarkably shares some hidden convexity. Hence, we devise a polynomial time procedure aimed at synthesizing an optimal MIMO waveform covariance. Finally, at the analysis stage, we assess the performance of the proposed design technique showing that it ensures an improved worst case performance than some counterparts available in open literature.

Quadratically constrained quadratic programming (QCQP) with double-sided constraints has plenty of applications in signal processing as have been addressed in recent years. QCQP problems are hard to solve, in general, and they are typically approached by solving a semidefinite programming (SDP) relaxation followed by a postprocessing procedure. Existing postprocessing schemes include Gaussian randomization to generate an approximate solution, rank reduction procedure (the so-called purification), and some specific rank-one matrix decomposition techniques to yield a globally optimal solution. In this paper, we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double-sided QCQP (i.e., instances with a small number of constraints). We illustrate their applicability in robust receive beamforming, radar optimal code design, and broadcast beamforming for multiuser communications. As a byproduct, we derive an alternative (shorter) proof for the Sturm-Zhang rank-one matrix decomposition theorem.

Quadratically constrained quadratic programming (QCQP) with double-sided constraints has plenty of applications in signal processing as have been addressed in recent years. QCQP problems are hard to solve, in general, and they are typically approached by solving a semidefinite programming (SDP) relaxation followed by a postprocessing procedure. Existing postprocessing schemes include Gaussian randomization to generate an approximate solution, rank reduction procedure (the so-called purification), and some specific rank-one matrix decomposition techniques to yield a globally optimal solution. In this paper, we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double-sided QCQP (i.e., instances with a small number of constraints). We illustrate their applicability in robust receive beamforming, radar optimal code design, and broadcast beamforming for multiuser communications. As a byproduct, we derive an alternative (shorter) proof for the Sturm-Zhang rank-one matrix decomposition theorem.

Consider a unicast downlink beamforming optimization problem with robust signal-to-interference-plus-noise ratio constraints to account for imperfect channel state information at the base station in a multiple-input single-output (MISO) communication system. The convexity of this robust beamforming problem remains unknown. A slightly conservative version of the robust beamforming problem is thus studied herein as a compromise. It is in the form of a semi-infinite second-order cone program (SOCP) and, more importantly, it possesses an equivalent and explicit convex reformulation, due to a linear matrix inequality (LMI) description of the cone of Lorentz-positive maps. Hence, the conservative robust beamforming problem can be efficiently solved by an optimization solver. Additional robust shaping constraints can also be easily handled to control the amount of interference generated on other co-existing users such as in cognitive radio systems.

Consider a unicast downlink beamforming optimization problem with robust signal-to-interference-plus-noise ratio constraints to account for imperfect channel state information at the base station in a multiple-input single-output (MISO) communication system. The convexity of this robust beamforming problem remains unknown. A slightly conservative version of the robust beamforming problem is thus studied herein as a compromise. It is in the form of a semi-infinite second-order cone program (SOCP) and, more importantly, it possesses an equivalent and explicit convex reformulation, due to a linear matrix inequality (LMI) description of the cone of Lorentz-positive maps. Hence, the conservative robust beamforming problem can be efficiently solved by an optimization solver. Additional robust shaping constraints can also be easily handled to control the amount of interference generated on other co-existing users such as in cognitive radio systems.

Consider a unicast downlink beamforming optimization problem with robust signal-to-interference-plus-noise ratio constraints to account for non-perfect channel state information at the base station. The convexity of the robust beamforming problem remains unknown. A slightly conservative version of the robust beamforming problem is thus studied herein as a compromise. It is in the form of a semi-infinite second-order cone program (SOCP), and more importantly, it possesses an equivalent and explicit convex reformulation, due to an linear matrix inequality description of the cone of Lorentz-positive maps. Hence the robust beamforming problem can be efficiently solved by an optimization solver. The simulation results show that the conservativeness of the robust form of semi-infinite SOCP is appropriate in terms of problem feasibility rate and the average transmission power.

Consider a unicast downlink beamforming optimization problem with robust signal-to-interference-plus-noise ratio constraints to account for non-perfect channel state information at the base station. The convexity of the robust beamforming problem remains unknown. A slightly conservative version of the robust beamforming problem is thus studied herein as a compromise. It is in the form of a semi-infinite second-order cone program (SOCP), and more importantly, it possesses an equivalent and explicit convex reformulation, due to an linear matrix inequality description of the cone of Lorentz-positive maps. Hence the robust beamforming problem can be efficiently solved by an optimization solver. The simulation results show that the conservativeness of the robust form of semi-infinite SOCP is appropriate in terms of problem feasibility rate and the average transmission power.

Two-way relaying can improve spectral efficiency in two-user cooperative communications. It also has great potential in multiuser systems. A major problem of designing a multiuser two-way relay system (MU-TWRS) is transceiver or precoding design to suppress co-channel interference. This paper aims to study linear precoding designs for a cellular MU-TWRS where a multi-antenna base station (BS) conducts bi-directional communications with multiple mobile stations (MSs) via a multi-antenna relay station (RS) with amplify-and-forward relay strategy. The design goal is to optimize uplink performance, including total mean-square error (Total-MSE) and sum rate, while maintaining individual signal-to-interference-plus-noise ratio (SINR) requirement for downlink signals. We show that the BS precoding design with the RS precoder fixed can be converted to a standard second order cone programming (SOCP) and the optimal solution is obtained efficiently. The RS precoding design with the BS precoder fixed, on the other hand, is non-convex and we present an iterative algorithm to find a local optimal solution. Then, the joint BS-RS precoding is obtained by solving the BS precoding and the RS precoding alternately. Comprehensive simulation is conducted to demonstrate the effectiveness of the proposed precoding designs., Comment: 13 pages, 12 figures, Accepted by IEEE TWC

This paper deals with the problem of estimating the steering direction of a signal, embedded in Gaussian disturbance, under a general quadratic inequality constraint, representing the uncertainty region of the steering. We resort to the maximum likelihood (ML) criterion and focus on two scenarios. The former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot. The latter supposes that the useful signal keeps a constant amplitude within all the snapshots. We prove that the ML criterion leads in both cases to a fractional quadratically constrained quadratic problem (QCQP). In order to solve it, we first relax the problem into a constrained fractional semidefinite programming (SDP) problem which is shown equivalent, via the Charnes-Cooper transformation, to an SDP problem. Then, exploiting a suitable rank-one decomposition, we show that the SDP relaxation is tight and give a procedure to construct (in polynomial time) an optimal solution of the original problem from an optimal solution of the fractional SDP. We also assess the quality of the derived estimator through a comparison between its performance and the constrained Cramer Rao lower Bound (CRB). Finally, we give two applications of the proposed theoretical framework in the context of radar detection.

This paper deals with the problem of estimating the steering direction of a signal, embedded in Gaussian disturbance, under a general quadratic inequality constraint, representing the uncertainty region of the steering. We resort to the maximum likelihood (ML) criterion and focus on two scenarios. The former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot. The latter supposes that the useful signal keeps a constant amplitude within all the snapshots. We prove that the ML criterion leads in both cases to a fractional quadratically constrained quadratic problem (QCQP). In order to solve it, we first relax the problem into a constrained fractional semidefinite programming (SDP) problem which is shown equivalent, via the Charnes-Cooper transformation, to an SDP problem. Then, exploiting a suitable rank-one decomposition, we show that the SDP relaxation is tight and give a procedure to construct (in polynomial time) an optimal solution of the original problem from an optimal solution of the fractional SDP. We also assess the quality of the derived estimator through a comparison between its performance and the constrained Cramer Rao lower Bound (CRB). Finally, we give two applications of the proposed theoretical framework in the context of radar detection.

This paper deals with the problem of estimating the steering direction of a signal, embedded in Gaussian disturbance, under a general quadratic inequality constraint, representing the uncertainty region of the steering. We resort to the maximum likelihood (ML) criterion and focus on two scenarios. The former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot. The latter supposes that the useful signal keeps a constant amplitude within all the snapshots. We prove that the ML criterion leads in both cases to a fractional quadratically constrained quadratic problem (QCQP). In order to solve it, we first relax the problem into a constrained fractional semidefinite programming (SDP) problem which is shown equivalent, via the Charnes-Cooper transformation, to an SDP problem. Then, exploiting a suitable rank-one decomposition, we show that the SDP relaxation is tight and give a procedure to construct (in polynomial time) an optimal solution of the original problem from an optimal solution of the fractional SDP. We also assess the quality of the derived estimator through a comparison between its performance and the constrained Cramer Rao lower Bound (CRB). Finally, we give two applications of the proposed theoretical framework in the context of radar detection.

Consider a downlink communication system where multiantenna base stations transmit independent data streams to decentralized single-antenna users over a common frequency band. The goal of the base stations is to jointly adjust the beamforming vectors to minimize the transmission powers while ensuring the signal-to-interference-noise ratio requirement of each user within the system. At the same time, it may be necessary to keep the interference generated on other coexisting systems under a certain tolerable level. In addition, one may want to include general individual shaping constraints on the beamforming vectors. This beamforming problem is a separable homogeneous quadratically constrained quadratic program, and it is difficult to solve in general. In this paper, we give conditions under which strong duality holds and propose efficient algorithms for the optimal beamforming problem. First, we study rank-constrained solutions of general separable semidefinite programs (SDPs) and propose rank reduction procedures to achieve a lower rank solution. Then we show that the SDP relaxation of three classes of optimal beamforming problem always has a rank-one solution, which can be obtained by invoking the rank reduction procedures.

This paper considers the downlink beamforming optimization problem that minimizes the total transmission power subject to global shaping constraints and individual shaping constraints, in addition to the constraints of quality of service (QoS) measured by signal-to-interference-plus-noise ratio (SINR). This beamforming problem is a separable homogeneous quadratically constrained quadratic program (QCQP), which is difficult to solve in general. Herein we propose efficient algorithms for the problem consisting of two main steps: 1) solving the semidefinite programming (SDP) relaxed problem, and 2) formulating a linear program (LP) and solving the LP (with closed-form solution) to find a rank-one optimal solution of the SDP relaxation. Accordingly, the corresponding optimal beamforming problem (OBP) is proven to be "hidden" convex, namely, strong duality holds true under certain mild conditions. In contrast to the existing algorithms based on either the rank reduction steps (the purification process) or the Perron-Frobenius theorem, the proposed algorithms are based on the linear program strong duality theorem.

Consider the downlink beamforming optimization problem with signal-to-interference-plus-noise ratio constraints, null-shaping interference constraints and multiple groups of individual shaping constraints. We propose an efficient algorithm for the problem, which consists of firstly solving the dual of the semidefinite programm (SDP) relaxation, secondly formulating a linear program (LP) and solving it to find a rankone solution of the SDP relaxation. In contrast to the existing algorithms, the analysis of the proposed algorithm includes neither the rank reduction steps (purification process) nor the Perron-Frobenius theorem. ©2010 IEEE.

Consider a downlink communication system where multiantenna base stations transmit independent data streams to decentralized single-antenna users over a common frequency band. The goal of the base stations is to jointly adjust the beamforming vectors to minimize the transmission powers while ensuring the signal-to-interference-noise ratio requirement of each user within the system. At the same time, it may be necessary to keep the interference generated on other coexisting systems under a certain tolerable level. In addition, one may want to include general individual shaping constraints on the beamforming vectors. This beamforming problem is a separable homogeneous quadratically constrained quadratic program, and it is difficult to solve in general. In this paper, we give conditions under which strong duality holds and propose efficient algorithms for the optimal beamforming problem. First, we study rank-constrained solutions of general separable semidefinite programs (SDPs) and propose rank reduction procedures to achieve a lower rank solution. Then we show that the SDP relaxation of three classes of optimal beamforming problem always has a rank-one solution, which can be obtained by invoking the rank reduction procedures.

This paper considers the downlink beamforming optimization problem that minimizes the total transmission power subject to global shaping constraints and individual shaping constraints, in addition to the constraints of quality of service (QoS) measured by signal-to-interference-plus-noise ratio (SINR). This beamforming problem is a separable homogeneous quadratically constrained quadratic program (QCQP), which is difficult to solve in general. Herein we propose efficient algorithms for the problem consisting of two main steps: 1) solving the semidefinite programming (SDP) relaxed problem, and 2) formulating a linear program (LP) and solving the LP (with closed-form solution) to find a rank-one optimal solution of the SDP relaxation. Accordingly, the corresponding optimal beamforming problem (OBP) is proven to be "hidden" convex, namely, strong duality holds true under certain mild conditions. In contrast to the existing algorithms based on either the rank reduction steps (the purification process) or the Perron-Frobenius theorem, the proposed algorithms are based on the linear program strong duality theorem.

This paper considers the downlink beamforming optimization problem that minimizes the total transmission power subject to global shaping constraints and individual shaping constraints, in addition to the constraints of quality of service (QoS) measured by signal-to-interference-plus-noise ratio (SINR). This beamforming problem is a separable homogeneous quadratically constrained quadratic program (QCQP), which is difficult to solve in general. Herein we propose efficient algorithms for the problem consisting of two main steps: 1) solving the semidefinite programming (SDP) relaxed problem, and 2) formulating a linear program (LP) and solving the LP (with closed-form solution) to find a rank-one optimal solution of the SDP relaxation. Accordingly, the corresponding optimal beamforming problem (OBP) is proven to be "hidden" convex, namely, strong duality holds true under certain mild conditions. In contrast to the existing algorithms based on either the rank reduction steps (the purification process) or the Perron-Frobenius theorem, the proposed algorithms are based on the linear program strong duality theorem.

Consider a downlink communication system where multiantenna base stations transmit independent data streams to decentralized single-antenna users over a common frequency band. The goal of the base stations is to jointly adjust the beamforming vectors to minimize the transmission powers while ensuring the signal-to-interference-noise ratio requirement of each user within the system. At the same time, it may be necessary to keep the interference generated on other coexisting systems under a certain tolerable level. In addition, one may want to include general individual shaping constraints on the beamforming vectors. This beamforming problem is a separable homogeneous quadratically constrained quadratic program, and it is difficult to solve in general. In this paper, we give conditions under which strong duality holds and propose efficient algorithms for the optimal beamforming problem. First, we study rank-constrained solutions of general separable semidefinite programs (SDPs) and propose rank reduction procedures to achieve a lower rank solution. Then we show that the SDP relaxation of three classes of optimal beamforming problem always has a rank-one solution, which can be obtained by invoking the rank reduction procedures.

Consider the downlink beamforming optimization problem with signal-to-interference-plus-noise ratio constraints, null-shaping interference constraints and multiple groups of individual shaping constraints. We propose an efficient algorithm for the problem, which consists of firstly solving the dual of the semidefinite programm (SDP) relaxation, secondly formulating a linear program (LP) and solving it to find a rankone solution of the SDP relaxation. In contrast to the existing algorithms, the analysis of the proposed algorithm includes neither the rank reduction steps (purification process) nor the Perron-Frobenius theorem. ©2010 IEEE.

Consider a downlink communication system where multi-antenna base stations transmit independent data streams to decentralized single-antenna users over a common frequency band. The goal of the base stations is to jointly adjust the beamforming vectors so as to minimize the transmission powers while ensuring the signal-to-interference-noise ratio (SINR) requirement of individual users within the system, and keeping lower interference level to other systems which operate in the same frequency band and in the same region. This optimal beamforming problem is a separable homogeneous quadratically constrained quadratical programming (QCQP), and it is difficult to solve in general. In this paper, we give conditions under which strong duality holds, and propose an efficient algorithm for the optimal beamforming problem. First, we study rank-constrained solutions of a general separable semidefinite programming (SDP), and propose a rank reduction procedure to achieve a lower rank solution. Then we show that the SDP relaxation of a class of the optimal beamforming problem has a rank-one solution, which can be obtained by invoking the rank reduction procedure.

Consider a downlink communication system where multi-antenna base stations transmit independent data streams to decentralized single-antenna users over a common frequency band. The goal of the base stations is to jointly adjust the beamforming vectors so as to minimize the transmission powers while ensuring the signal-to-interference-noise ratio (SINR) requirement of individual users within the system, and keeping lower interference level to other systems which operate in the same frequency band and in the same region. This optimal beamforming problem is a separable homogeneous quadratically constrained quadratical programming (QCQP), and it is difficult to solve in general. In this paper, we give conditions under which strong duality holds, and propose an efficient algorithm for the optimal beamforming problem. First, we study rank-constrained solutions of a general separable semidefinite programming (SDP), and propose a rank reduction procedure to achieve a lower rank solution. Then we show that the SDP relaxation of a class of the optimal beamforming problem has a rank-one solution, which can be obtained by invoking the rank reduction procedure.

Finally, as combinatorial applications of complex quadratic optimization; we consider Max 3-Cut with fixed nodes constraints, Max 3-Dicut with weight constraints, Max 3-XOR, and so on, and present corresponding bounds on the approximation ratios., Quadratic optimization problems with complex-valued decision variables, in short called complex quadratic optimization problems, find many applications in engineering. In this dissertation, we study several instructive models of complex quadratic optimization, as well as its applications in combinatorial optimization. The tool that we use is a combination of semidefinite programming (SDP) relaxation and randomization technique, which has been well exploited in the last decade. Since most of the optimization models are NP-hard in nature, we shall design polynomial time approximation algorithms for a general model, or polynomial time exact algorithms for some restricted instances of a general model., To enable the analysis, we first develop two basic theoretical results: one is the probability formula for a bivariate complex normal distribution vector to be in a prescribed angular region, and the other one is the rank-one decomposition theorem for complex positive semidefinite matrices. The probability formula enables us to compute the expected value of a randomized (with a specific rounding rule) solution based on an optimal solution of the SDP relaxation problem, while the rank-one decomposition theorem provides a new proof of the complex S -lemma and leads to novel deterministic rounding procedures., With the above results in hand, we then investigate the models and applications of complex quadratic optimization via semidefinite programming in detail. We present an approximation algorithm for a convex quadratic maximization problem with discrete complex decision variables, where the approximation analysis is based on the probability formula. Besides, an approximation algorithm is proposed for a non-convex quadratic maximization problem with discrete complex decision variables. Then we study a limit of the model, i.e., a quadratic maximization problem with continuous unit module decision variables. The problem is shown to be strongly NP-hard. Approximation algorithms are described for the problem, including both convex case and non-convex case. Furthermore, if the objective matrix has a sign structure, then a stronger approximation result is shown to hold. In addition, we use the complex matrix decomposition theorem to solve complex quadratically constrained complex quadratic programming. We consider several interesting cases for which the corresponding SDP relaxation admits no gap with the true optimal value, and consequently, this yields polynomial time procedures for solving those special cases of complex quadratic optimization., Huang Yongwei., "August 2005.", Adviser: Shuzhong Zhang., Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 4033., Thesis (Ph.D.)--Chinese University of Hong Kong, 2005., Includes bibliographical references (p. 142-155)., Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web., Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web., in English and Chinese., School code: 1307., isbn: 9780542760501, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Finally, as combinatorial applications of complex quadratic optimization; we consider Max 3-Cut with fixed nodes constraints, Max 3-Dicut with weight constraints, Max 3-XOR, and so on, and present corresponding bounds on the approximation ratios., Quadratic optimization problems with complex-valued decision variables, in short called complex quadratic optimization problems, find many applications in engineering. In this dissertation, we study several instructive models of complex quadratic optimization, as well as its applications in combinatorial optimization. The tool that we use is a combination of semidefinite programming (SDP) relaxation and randomization technique, which has been well exploited in the last decade. Since most of the optimization models are NP-hard in nature, we shall design polynomial time approximation algorithms for a general model, or polynomial time exact algorithms for some restricted instances of a general model., To enable the analysis, we first develop two basic theoretical results: one is the probability formula for a bivariate complex normal distribution vector to be in a prescribed angular region, and the other one is the rank-one decomposition theorem for complex positive semidefinite matrices. The probability formula enables us to compute the expected value of a randomized (with a specific rounding rule) solution based on an optimal solution of the SDP relaxation problem, while the rank-one decomposition theorem provides a new proof of the complex S -lemma and leads to novel deterministic rounding procedures., With the above results in hand, we then investigate the models and applications of complex quadratic optimization via semidefinite programming in detail. We present an approximation algorithm for a convex quadratic maximization problem with discrete complex decision variables, where the approximation analysis is based on the probability formula. Besides, an approximation algorithm is proposed for a non-convex quadratic maximization problem with discrete complex decision variables. Then we study a limit of the model, i.e., a quadratic maximization problem with continuous unit module decision variables. The problem is shown to be strongly NP-hard. Approximation algorithms are described for the problem, including both convex case and non-convex case. Furthermore, if the objective matrix has a sign structure, then a stronger approximation result is shown to hold. In addition, we use the complex matrix decomposition theorem to solve complex quadratically constrained complex quadratic programming. We consider several interesting cases for which the corresponding SDP relaxation admits no gap with the true optimal value, and consequently, this yields polynomial time procedures for solving those special cases of complex quadratic optimization., Huang Yongwei., "August 2005.", Adviser: Shuzhong Zhang., Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 4033., Thesis (Ph.D.)--Chinese University of Hong Kong, 2005., Includes bibliographical references (p. 142-155)., Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web., Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web., in English and Chinese., School code: 1307., isbn: 9780542760501, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)