matrix-monotonic optimization

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 69, 2021 179

Matrix-Monotonic Optimization − Part II:Multi-Variable Optimization

Chengwen Xing , Member, IEEE, Shuai Wang , Member, IEEE, Sheng Chen , Fellow, IEEE,Shaodan Ma , Member, IEEE, H. Vincent Poor , Fellow, IEEE, and Lajos Hanzo , Fellow, IEEE

Abstract—In contrast to Part I of this treatise (Xing, 2021)that focuses on the optimization problems associated with singlematrix variables, in this paper, we investigate the application ofthe matrix-monotonic optimization framework in the optimizationproblems associated with multiple matrix variables. It is revealedthat matrix-monotonic optimization still works even for multiplematrix-variate based optimization problems, provided that certainconditions are satisfied. Using this framework, the optimal struc-tures of the matrix variables can be derived and the associatedmultiple matrix-variate optimization problems can be substantiallysimplified. In this paper several specific examples are given, whichare essentially open problems. Firstly, we investigate multi-usermultiple-input multiple-output (MU-MIMO) uplink communica-tions under various power constraints. Using the proposed frame-work, the optimal structures of the precoding matrices at eachuser under various power constraints can be derived. Secondly, weconsidered the optimization of the signal compression matrices ateach sensor under various power constraints in distributed sensornetworks. Finally, we investigate the transceiver optimization formulti-hop amplify-and-forward (AF) MIMO relaying networkswith imperfect channel state information (CSI) under variouspower constraints. At the end of this paper, several simulationresults are given to demonstrate the accuracy of the proposedtheoretical results.

Index Terms—MIMO, matrix-monotonic optimization, multiplematrix-variate optimizations.

Manuscript received May 5, 2020; revised August 27, 2020 and September 29,2020; accepted October 22, 2020. Date of publication November 11, 2020; dateof current version February 3, 2021. The associate editor coordinating the reviewof this article and approving it for publication was Prof. Stefano Tomasin. Thework of Chengwen Xing was supported in part by the National Natural ScienceFoundation of China under Grants 61671058, 61722104, and 61620106001,and in part by Ericsson. The work of Shaodan Ma was partially supportedby the Science and Technology Development Fund, Macau SAR (File no.0036/2019/A1 and File no. SKL-IOTSC2018-2020), and in part by the ResearchCommittee of University of Macau under Grant MYRG2018-00156-FST. Thework of H. Vincent Poor was supported by the U.S. National Science Foundationunder Grant CCF-1908308. (Corresponding author: Shuai Wang.)

Chengwen Xing is with the School of Information and Electronics, BeijingInstitute of Technology, Beijing 100081, China, and also with the Departmentof Electrical and Computer Engineering, University of Macau, Macao S.A.R.999078, China (e-mail: [email protected]).

Shuai Wang is with the School of Information and Electronics, BeijingInstitute of Technology, Beijing 100081, China (e-mail: [email protected]).

Sheng Chen is with the School of Electronics and Computer Science, Univer-sity of Southampton, Southampton 430205, U.K., and also with the King Ab-dulaziz University, Jeddah 21422, Saudi Arabia (e-mail: [email protected]).

Shaodan Ma is with the State Key Laboratory of Internet of Things for SmartCity, Department of Electrical and Computer Engineering, University of Macau,Macao, Macao S.A.R. 999078, China (e-mail: [email protected]).

H. Vincent Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: [email protected]).

Lajos Hanzo is with the School of Electronics and Computer Science, Univer-sity of Southampton, Southampton 430205, U.K. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2020.3037495

I. MOTIVATIONS

THE deployment of multi-antenna arrays opened a doorto effectively exploit spatial resources to improve energy

efficiency and spectrum efficiency [1]–[5]. Meanwhile, the in-volved design variables are usually matrices instead of simplescalars [6]–[8]. In order to solve the matrix-variate optimiza-tion problems for MIMO communications efficiently, the mostwidely used logic is first to derive the optimal structures ofthe matrix variables. Then based on the optimal structures,the considered optimization problems can be greatly simplified[9]–[16].

Matrix-monotonic optimization is an interesting frameworkthat takes advantage of monotonic property in positive semidef-inite matrix set to derive the optimal structures of optimizationvariables [1], [17]–[20]. In Part I [1], we focus our attention onsingle-variable optimization problems. However, for many prac-tical optimization problems there are multiple matrix variates tooptimize. For example, in multi-user multiple-input multiple-output (MU-MIMO) communication systems, the transceiveroptimization processes of the downlink and uplink involve multi-ple matrix variables, namely the equalizer matrices and precodermatrices [21]–[24]. For multi-carrier MIMO systems, in eachsubcarrier there is a precoder matrix and an equalizer matrix [17].Moreover, in multi-hop communications the forwarding matrixof each relay has to be optimized [25], [26].

This fact inspires us to take a further step and to inves-tigate the optimization problems hinging on multiple matrix-variables. Generally speaking, solving an optimization problemhaving multiple matrix-variables is more challenging than itssingle matrix-variable counterpart. How to solve this kind ofoptimization problems has attracted substantial attention bothacross the wireless communication and signal processing re-search communities [21]–[23]. In contrast to single matrix-variable optimizations, for multiple matrix-variable optimiza-tion in most cases it is impossible to derive the optimal solutionsin closed-form. Iterative optimization algorithms or alternatingoptimization algorithms have neeb widely used to solve thiskind of optimization problems [23]–[27]. Unfortunately, thereis no general-purpose mathematical tool or framework that cancover all the kinds of optimization problems. In some cases,similar to the single-variable case, for multiple matrix-variableoptimization first the optimal structures of the matrix variableshave to be derived, based on which the optimization can besignificantly simplified and the corresponding convergence ratecan be substantially improved.

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180 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 69, 2021

In this paper, we investigate in detail, how to exploit thehidden monotonicity in positive semidefinite matrix fields toderive the optimal structures of the multiple matrix variables.Based on the optimal structures, the optimizations of multiplematrix variables can be significantly simplified. In our work, itis revealed that for many optimization problems associated withmultiple matrix variables, the matrix-monotonic optimizationframework still works. We also would like to point out that theauthors of [17] also investigate how to apply matrix-monotonicoptimization to optimization problems associated with multiplematrix variables. However, it is worth highlighting that theprevious contribution [17] only considers a simple sum powerconstraint. By contrast, our work in this paper is significantlydifferent from that in [17], since here diverse power constraintsare taken into account, such as the multiple weighted powerconstraints of [20], the shaping constraint of [28], [29] and soon. Additionally, more scenarios are also taken into account.Furthermore, in addition to the multi-hop systems investigatedin [17], in this paper, the MU-MIMO uplink and distributedsensor networks are also considered.

The main contributions of this paper are enumerated in thefollowing. These contributions distinguish our work from theexisting related works.� Firstly, we investigate precoder optimization in the up-

link of MU-MIMO communications under three differentpower constraints, namely the shaping constraint, jointpower constraint and multiple weighted power constraints.Based on the matrix-monotonic optimization framework,the optimal structures of the matrix variables can be de-rived. Then the optimization can be substantially simplifiedand can be efficiently solved by an iterative algorithm.In each iteration based on the optimal structure, the op-timal solutions of the remaining variables are standardwater-filling solutions. We cover the precoder optimiza-tion under per-antenna power constraint as its specialcases.

� Secondly, we investigate the signal compression matrixoptimization problem in a distributed sensor network underthe above three power constraints. For this data fusionoptimization, there exist correlations between the signalstransmitted from different sensors. This makes the cor-responding optimization problem significantly differentfrom that in the MU-MIMO uplink. Moreover, in con-trast to [27], where at each sensor only the sum powerconstraint is considered, in our work more general powerconstraints are taken into account, namely the shapingconstraint, joint power constraint and multiple weightedpower constraints. This is our main contribution. Based onthe matrix-monotonic optimization framework, the optimalstructures of the compression matrices can be derived andthe optimal solutions of the remaining vectors are found tocorrespond to water-filling solutions.

� Thirdly, we investigate the robust transceiver optimizationproblem of multi-hop amplify-and-forward (AF) cooper-ative MIMO networks, including both linear and nonlin-ear transceivers. For the linear transceivers, various kindsof performance metrics are taken into account, namely

the additively Schur-convex and Schur-concave scenar-ios [25], [26], [30]. On the other hand, for nonlineartransceivers, both decision feedback equalizers (DFE) andTomlinson-Harashima precoding (THP) are investigated.In contrast to [28], [31], various power constraints aretaken into account in the robust transceiver optimizationinstead of the simple sum power constraint. Based on theproposed framework, the optimal structures of the robusttransceiver design can be derived, based on which therobust transceiver optimization can be efficiently solved.Hence our contribution fills a void in the robust transceiverdesign literature of multi-hop AF MIMO systems undermultiple weighted power constraints.

The remainder of this paper is organized as follows. InSection II, the basic properties of the framework on matrix-monotonic optimizations are given first. Following that, the MU-MIMO uplink is investigated in Section III. Compression matrixoptimization for distributed sensor networks is discussed basedon matrix-monotonic optimization in Section IV. In Section V,robust transceiver optimization is proposed for multi-hop AFMIMO relaying networks separately under shaping constraints,joint power constraints and multiple weighted power constraints.Several numerical results are given in Section VII, Finally, theconclusions are drawn in Section VIII.

Notation: To be consistent with our Part I work [1], thefollowing notations and definitions are used throughout thispaper. The symbols ZH, ZT, Tr(Z) and |Z| denote the Her-mitian transpose, transpose, trace and determinant of matrixZ, respectively. The matrix Z

12 is the Hermitian square root

of a positive semidefinite matrix Z, which is also a positivesemidefinite matrix. For anN ×N matrixZ, the vectorλ(Z) isdefined asλ(Z) = [λ1(Z), . . . , λN (Z)]T whereλi(Z) denotesthe ith largest eigenvalue of Z. The symbol [Z]i,j denotes theith-row and jth-column element. On the other hand, the symbold(Z) denotes the vector consisting of the diagonal elements ofZ. The identity matrix is denoted by I . In this paper, Λ alwaysrepresents a diagonal matrix, and Λ ↘ and Λ ↗ represent arectangular or square diagonal matrix with the diagonal elementsin descending order and ascending order, respectively.

II. FUNDAMENTALS OF MATRIX-MONOTONIC OPTIMIZATION

In this paper, we investigate a real valued optimization prob-lem with multiple complex matrix variables {Xk}Kk=1 which isgenerally formulated as

Opt.1.1 : min{Xk}Kk=1

f0({Xk}Kk=1),

s.t. ψk,i(Xk) ≤ 0,

1 ≤ k ≤ K, 1 ≤ i ≤ Ik, (1)

where ψk,i(·), 1 ≤ k ≤ K, 1 ≤ i ≤ Ik, denotes the constraintfunctions. Similar to the single-variate matrix-monotonic opti-mization investigated in Part I [1], all constraints considered inthis paper are right unitarily invariant, i.e., for arbitrary unitarymatrices QXk

’s,

ψk,i

(XkQXk

)= ψk,i (Xk) . (2)


XING et al.: MATRIX-MONOTONIC OPTIMIZATION − PART II: MULTI-VARIABLE OPTIMIZATION 181

In the following, several specific power constraints are given.The general power constraint model is one of the main contri-butions of this work.

A. The Constraints on Multiple Matrix Variables

The simplest constraint is sum power constraint, i.e., the sumpower across all transmit antennas is smaller than a predefinedthreshold. For example, in MU-MIMO uplink communications,each mobile terminal has a sum power constraint such as

Tr(XkX

Hk

) ≤ Pk. (3)

It is obvious that the sum power constraint is right unitarilyinvariant. Moreover, in order to constrain the fluctuation of theeigenvalues ofXkX

Hk , the following joint power constraint will

be used [28], [29]

Tr(XkX

Hk

) ≤ Pk, XkXHk � τkI. (4)

The difference between the sum power constraint and the jointpower constraint is that there is an additional maximum eigen-value constraint. It is obvious that the joint power constraint isright unitarily invariant.

From the circuit viewpoint, each amplifier is connected toone distinct antenna. It is not reasonable to use the sum poweras a constraint as the powers cannot be shared between differentantennas. In other words, the individual power constraint or per-antenna power constraint is more practical, which is formulatedas [21], [31], [32][

XkXHk

]n,n

≤ Pk,n, n = 1, . . . , N. (5)

The per-antenna power constraint is also right unitarily invariant.It is worth highlighting that the per-antenna power constraintcannot include the sum power constraint as its special case.

In order to build a more general constraint model includingmore specific power constraints as its special cases, multipleweighted power constraints are given in the following [1], [20]

Tr(Ωk,iXkX

Hk

) ≤ Pk,i, i = 1, . . . , Ik, (6)

where Ik is the number of weighted power constraints for thekth variable Xk. The positive semidefinite matrices Ωk,i’s areweighting matrices. The multiple weighted power constraintsare right unitarily invariant as well.

Finally, in order to constrain the transmit signals to be in a de-sired region, shaping constraint can be used. Shaping constraintis a constraint on the covariance matrix of transmitted signals.Specifically, the shaping constraint on a matrix variable Xk isdefined as [28], [33]

XkXHk � Rsk , (7)

where Rsk is the desired signal shaping matrix [28], [33]. Theshaping constraint (7) is right unitarily invariant as well. Underthese power constraints, in the following we give the propertieswhich are the basis of application of the framework of matrix-monotonic optimization.

From a mathematical perspective, the more complicatedpower constraints will significantly change the feasible set com-pared to that of the sum power constraint. This is because the sum

power constraint is both right unitarily invariant and left unitarilyinvariant, however the general power constraints are only rightunitarily invariant. In other words, the symmetry of sum powerconstraint does not exist for the general power constraints suchas multiple weighted power constraints. It is clear that undermultiple weighted power constraints the extreme values and theoptimal solutions are significantly different from that under thesum power constraint. The multiple weighted power constraintsmodel also includes the sum power constraint model as itsspecial case. Note that the sum power constraint model is nota special case of the per-antenna power constraint model. Onemodel can include two different constraint models as its specialcases. This is also an advantage of the multiple weighted powerconstraints model.

B. Matrix-Monotonic Properties

The framework of matrix-monotonic optimization aims at ex-ploiting the monotonicity in positive semidefinite field to derivethe optimal structures of the matrix variates. As the constraintsin Opt. 1.1 are right unitarily invariant, definingXk = F kQXk

Opt. 1.1 is equivalent to the following optimization problem

Opt.1.2 : min{F k,QXk

}Kk=1

f0({F kQXk}Kk=1),

s.t. ψk,i(F k) ≤ 0,

1 ≤ k ≤ K, 1 ≤ i ≤ Ik (8)

In our work, Opt. 1.2: satisfies the following properties.For the kth optimal unitary matrix QXk

, the objectivefunction in Opt. 1.2 can be transferred into a function ofλ(FH

k ΠkF k) i.e.,

f0({F kQXk}Kk=1) = g0,k(λ(F

Hk ΠkF k)), for k = 1, . . . ,K

(9)

with g0,k(λ(FHk ΠkF k)) being a monotonically decreasing

function with respect to λ(FHk ΠkF k) for k = 1, . . . ,K. The

optimal F k is a Pareto optimal solution of the following vectoroptimization subproblem

Opt.1.3 : maxF k

λ(FHk ΠkF k),

s.t. ψk,i(F k) ≤ 0, 1 ≤ i ≤ Ik, (10)

which is equivalent to the following matrix-monotonic optimiza-tion problem [1], [17]

Opt.1.4 : maxF k

FHk ΠkF k,

s.t. ψk,i(F k) ≤ 0, 1 ≤ i ≤ Ik. (11)

where Πk is independent of F k. Then, based on the results ofPart I [1], the optimal structure of F k can be derived. Based onthe optimal structures, the optimization problem can be substan-tially simplified. To elaborate a little further, given the optimalstructures, the optimization problem Opt. 1.2 associated withmultiple matrix variables can be efficiently solved in an iterativemanner. It is worth noting that given the optimal structures,an iterative algorithm is still needed to solve Opt. 1.2 and in



most cases the iterative algorithms used are iterative water-fillingalgorithms [53], [54]. Suffice to say that the convergence of thiskind of algorithms can be guaranteed, but in a general case afterconvergence only covergence to the local optimum of the finalsolutions can be guaranteed. Based on Part I [1], in the followingthe fundamental results for Opt. 1.4 are given, which constitutethe basis for the following sections.

Shaping Constraint For the shaping constraint, Opt. 1.4becomes the following optimization problem [28]

Opt.1.5 : maxF kFH

k ΠkF k

s.t. F kFHk � Rsk . (12)

The following lemma reveals the optimal structure of F k forOpt. 1.5 with the shaping constraint.

Lemma 1: When the rank ofRsk is not higher than the numberof columns and the number of rows in F k, the optimal solutionF opt,k of Opt. 1.5 is a square root of Rsk , i.e., F opt,kF

Hopt,k =

Rsk .Joint Power Constraint Under the joint power constraint,

Opt. 1.4 can be rewritten as

Opt.1.6 : maxF kFH

k ΠkF k

s.t. Tr(F kF

Hk

)≤Pk,F kFHk � τkI. (13)

The Pareto optimal solution F opt,k for Opt. 1.6 is given inLemma 2.

Lemma 2: For Opt. 1.6 with the joint power constraint, thePareto optimal solutions satisfy the following structure

F opt,k = UΠkΛF k

UHArb,k, (14)

where the unitary matrix UΠkis specified by the EVD

Πk = UΠkΛΠk

UHΠk

with ΛΠk↘, (15)

every diagonal element of the rectangular diagonal matrix ΛF k

is smaller than√τk, and UArb,k is an arbitrary unitary matrix

having the appropriate dimension.Multiple Weighted Power Constraints Under the multiple

weighted power constraints, Opt. 1.4 becomes

Opt.1.7 : maxF k

FHk ΠkF k

s.t. Tr(Ωk,iF kF

Hk

)≤Pk,i, 1 ≤ i ≤ Ik. (16)

Note that the weighted power constraints include both the sumpower constraint and per-antenna power constraints as its specialcases. The Pareto optimal solution F opt,k for Opt. 1.7 is givenin Lemma 3.

Lemma 3: The Pareto optimal solutions of Opt. 1.6 satisfythe following structure

F opt,k = Ω− 1

2

k U˜Πk

Λ˜F k

UHArb,k, (17)

where UArb,k is an arbitrary unitary matrix of appropri-ate dimension, Ωk =

∑Iki=1 αk,iΩk,i, the nonnegative scalars

αk,i are the weighting factors that ensure that the constraintsTr(Ωk,iF kF

Hk ) ≤ Pk,i hold and they can be computed by

classic subgradient methods, while the unitary matrix U˜Πk

is

specified by the EVD

Ω− 1

2

k ΠkΩ− 1

2

k = U˜Πk

Λ˜Πk

UH˜Πk

with Λ˜Πk

↘ . (18)

In this paper, we focus our attention on the optimization prob-lems of multiple complex matrix variates. In order to overcomethe difficulties arising from the coupling relationships among themultiple matrix variates, the right unitarily invariant property ofthe constraints in Opt. 1.1 is exploited to introduce a series ofauxiliary unitary matrices. Each auxiliary unitary matrix alignsits corresponding matrix variable to achieve extreme objectivevalues. As a result, the optimal solutions of the matrix variablesare Pareto optimal solutions of a series of single-variate matrixmonotonic optimization problems. Then the optimal structure ofeach matrix variable can be derived, based on which the originaloptimization problem can be solved efficiently in an iterativemanner. In the following, three specific optimization problemswill be investigated, namely transceiver optimization for themulti-user MIMO (MU-MIMO) uplink, signal compression fordistributed sensor networks and transceiver optimizations formulti-hop amplify-and-forward (AF) MIMO relaying networks.Generally speaking, an auxiliary unitary matrix aligns its left-hand side and righthand side with its corresponding matrixvariables. The three examples are specifically chosen for char-acterizing the effects of the matrix variates on the auxiliaryunitary matrices. Specifically, in the transceiver optimization ofthe MU-MIMO uplink, when optimizing the kth matrix variate,the other matrix variates only affect the righthand side of thecorresponding unitary matrix. As for signal compression indistributed sensor networks, when optimizing the kth matrixvariate, the effects of other matrix variates are only on thelefthand side of the corresponding unitary matrix. Finally, asfor transceiver optimizations in AF MIMO relaying networks,when optimizing the kth matrix variate, the other matrix variatesaffect both sides of the corresponding unitary matrix.

III. MU-MIMO UPLINK COMMUNICATIONS

The first application scenario for the matrix monotonic opti-mization theory is found in MU MIMO uplink communications.In the MU MIMO uplink system of Fig. 1,Kmulti-antenna aidedmobile users communicate with a multi-antenna assisted basestation (BS) [34]–[37]. The BS recovers the signals transmittedfrom all the K mobile terminals. The sum rate maximizationproblem associated with this MU-MIMO uplink can be formu-lated as follows [21], [34]–[36]

Opt.2.1: min{P k}Kk=1

− log

∣∣∣∣Rn +K∑

k=1

HkXkW kXHk H

Hk

∣∣∣∣,s.t. ψk,i(Xk) ≤ 0, 1 ≤ i ≤ Ik, 1 ≤ k ≤ K,

(19)

where Hk is the MIMO channel matrix between the kth userand the BS, Xk is the precoding matrix at the kth user, andRn is the additive noise’s covariance matrix at the BS. For thekth user, the positive definite matrix W k is the correspondingweighting matrix. Different from the work in [21], the powerconstraints considered in our work are more general than the per-antenna power constraints in [21]. Similar to Opt. 1.2, defining



Fig. 1. The uplink of MU-MIMO communications.

Xk = F kQXk, the optimization problem (19) is equivalent to

Opt.2.2: min{F k}Kk=1

− log

∣∣∣∣Rn+K∑

k=1

HkF kQXkW kQ

HXk

FHk H

Hk

∣∣∣∣,s.t. ψk,i(F k) ≤ 0, 1 ≤ i ≤ Ik, 1 ≤ k ≤ K,

(20)

The objective function of Opt. 2.2 satisfies the following prop-erty, which can be exploited to optimize the multiple matrixvariables

log

∣∣∣∣Rn +

K∑k=1

HkF kQXkW kQ

HXk

FHk H

Hk

∣∣∣∣= log

∣∣∣∣I +HkF kQXkW kQ

HXk

FHk H

Hk

×(Rn +

∑j �=k

HjF jQXkW jQ

HXk

FHj H

Hj

)−1∣∣∣∣+ log

∣∣∣∣Rn +∑j �=k

HjF jQXjW jQ

HXj

FHj H

Hj

∣∣∣∣= log

∣∣∣∣I +W kQHXk

FHk H

Hk K

−1nkHkF kQXk

∣∣∣∣+ log∣∣Knk

∣∣,(21)

where we have

Knk= Rn +

∑j �=k

HjF jQXjW jQ

HXj

FHj H

Hj . (22)

Therefore, based on (21) for the kth matrix variate F k

Opt. 2.2 can be written in the following equivalent formula

Opt.2.3: minF k

− log

∣∣∣∣I+W kQHXk

FHk H

Hk K

−1nkHkF kQXk

∣∣∣∣,s.t. Knk

=Rn+∑j �=k

HjF jQXjW jQ

HXj

FHj H

Hj ,

ψk,i(F k) ≤ 0, i = 1, . . . , Ik. (23)

The matrix FHk H

Hk K

−1nkHkF k can be interpreted as the matrix

version SNR for the kth user [31]. Based on Matrix Inequality

4 in Part I [1], we have

log

∣∣∣∣I +W kQHXk

FHk H

Hk K

−1nkHkF kQXk

∣∣∣∣≤∑i

log(1 + λi(W )λi(F

Hk H

Hk K

−1nkHkF k)

). (24)

The equality holds when the unitary matrix QXkequals

Qopt,Xk= USNR,kU

HW (25)

where the unitary matrices USNR,k and UW kare defined based

on the following EVDs

FHk H

Hk K

−1nkHkF k

= USNR,kΛSNR,kUHSNR,k with ΛSNR,k ↘

W k = UW kΛW k

UHW k

with ΛW k↘ . (26)

From the multi-objective optimization viewpoint, the optimalsolutions of Opt. 2.3 belong to the Pareto optimal solution setsof the following optimization problems for 1 ≤ k ≤ K

Opt.2.4 : maxF k

λ(FH

k HHk K

−1nkHkF k

),

s.t. Knk=Rn+

∑j �=k

HjF jQXjW jQ

HXj

FHj H

Hj ,

ψk,i(F k) ≤ 0, i = 1, . . . , Ik. (27)

which is equivalent to

Opt.2.5: maxF k

FHk H

Hk K

−1nkHkF k,

s.t. Knk=Rn+

∑j �=k

HjF jQXjW jQ

HXj

FHj H

Hj ,

ψk,i(F k) ≤ 0, i = 1, . . . , Ik. (28)

It can be seen that by using alternating optimization algorithm,the multiple-matrix-variate optimization of Opt. 2.3 is trans-ferred into the multiple single-matrix-variate matrix-monotonicoptimization of Opt. 2.5. Based on Opt. 2.5, the optimal struc-ture of F k can be derived, and then the original optimizationproblem Opt. 2.2 can be solved in an iterative manner. It isworth noting that in most cases, for the alternating optimizationalgorithm, the final solutions are suboptimal. The alternating op-timization algorithm stops when the performance improvementis smaller than a predefined threshold or the iteration numberreaches the predefined maximum value. The convergence canbe guaranteed when the subproblems are solved with globaloptimality.

1) Shaping Constraint: We have Ik = 1 and

ψk,1(F k) = F kFHk −Rsk . (29)

Based on Lemma 1 in Section II, we readily conclude that for1 ≤ k ≤ K, when the rank of Rsk is not higher than the numberof columns and the number of rows in F k, the optimal solutionF opt,k of Opt. 2.3 is a square root of Rsk .

2) Joint Power Constraint: We have Ik = 2 and

ψk,1(F k) = Tr(F kF

Hk

)− Pk,

ψk,2(F k) = F kFHk − τkI.

(30)

Based on Lemma 2 in Section II, we readily conclude that for1 ≤ k ≤ K, the optimal solution F opt,k of Opt. 2.3 satisfies the



Fig. 2. Illustration of distribute sensor network.

following structure

F opt,k = V˜Hk

ΛF kUH

Arb,k, (31)

where the unitary matrix V˜Hk

is defined based on the SVD

K− 1

2nk Hk = U

˜HkΛ

˜HkV H

˜Hkwith Λ

˜Hk↘ . (32)

and every diagonal element of the rectangular diagonal matrixΛF k

is smaller than√τk. The diagonal matrix ΛF k

can beefficiently solved using a variant water-filling algorithm [52],[54].

3) Multiple Weighted Power Constraints: In this case, we have

ψk,i(F k) = Tr(Ωk,iF kF

Hk

)− Pk,i. (33)

Then based on Lemma 3 in Section II, we conclude that for1 ≤ k ≤ K, the optimal solution F opt,k of Opt. 2.3 satisfiesthe following structure

F opt,k = Ω− 1

2

k V HkΛ

˜F kUH

Arb,k, (34)

where the unitary matrix V Hkis defined by the following SVD

K− 1

2nk HkΩ

− 12

k = UHkΛHk

V HHk

with ΛHk↘, (35)

and the matrix Ωk is defined as

Ωk =

Ik∑i=1

αk,iΩk,i. (36)

The diagonal matrix Λ˜F k

can be efficiently solved using water-filling algorithms [53].

IV. SIGNAL COMPRESSION FOR DISTRIBUTED

SENSOR NETWORKS

In the distributed sensor network illustrated in Fig. 2, theK sensors transmit their individual signals to the fusion cen-ter [38]–[47]. Specifically, the kth sensor transmits its signal xk

to the fusion center, when the channel between the kth sensorand the fusion center is Hk. The fusion center recovers thetransmitted signalsxk for 1 ≤ k ≤ K. In contrast to the scenarioof MU-MIMO communications, there exist correlations amongxk [27], and the correlation matrix is denoted by

Cx = E

{[xT1 · · ·xT

K

]T [xT1 · · ·xT

K

]∗}. (37)

Note that the correlations among the signals make the optimiza-tion approach of this application totally different from that ofthe MU-MIMO application.

To maximize the mutual information between the receivedsignal at the data fusion center and the signal to estimate, thesignal compression can be formulated as Opt. 3.1 [27], given as

Opt.3.1: min{Xk}Kk=1

− log∣∣∣C−1

x

+ diag{{

XHk H

Hk R

−1nkHkXk

}Kk=1

} ∣∣∣,s.t. ψk,i

(XkR

12xk

)≤0, 1≤ i≤Ik, 1≤k≤K,

(38)

where F k is the signal compression matrix at the kth sensor,Rxk

is the covariance matrix of the signal xk transmitted fromthe kth sensor, and Rnk

is the covariance matrix of the additivenoise nk for the kth sensor signal received in its own time slotat the fusion center. Note that if all the sensors send signals atthe same frequency, all the Rnk

are identical. If the sensors usedifferent frequency bands, the noise covariance matrices Rnk

are different.Note that in [27], only the simple sum power constraint

is considered, while in our work the more general multipleweighted linear power constraints are taken into account. In otherwords, the result derived in this section for signal compressionin distributed sensor networks is novel.

For the general correlation matrixCx, it is difficult to directlydecouple the optimization problem. A natural choice is to takeadvantage of alternating optimization algorithms among Xk for1 ≤ k ≤ K. To simplify the derivation, a permutation matrixP k is first introduced, which reorders the block diagonal matrixdiag{{XH

k HHk R

−1nkHkXk}Kk=1} so that the following equality

holds

P kdiag{{

XHk H

Hk R

−1nkHkXk

}Kk=1

}PH

k

=

[XH

k HHk R

−1nkHkXk 0

0 Ξk

]. (39)

The computation of P k and the definition of Ξk are providedin Appendix A. The permutation matrix P k aims at movingthe term XH

k HHk R

−1nkHkXk at the top of the block diagonal

matrix. The permutation matrixP k is determined by the positionof the term XH

k HHk R

−1nkHkXk in the block diagonal matrix

diag{{XHk H

Hk R

−1nkHkXk}Kk=1}. Note that a permutation ma-

trix is also a unitary matrix. By further exploiting the propertiesof matrix determinants, Opt. 3.1 becomes equivalent to Opt. 3.2of (40).

Opt.3.2 : min{F k}Kk=1

− log∣∣∣P kC

−1x PH

k

+P kdiag{{

XHk H

Hk R

−1nkHkXk

}Kk=1

}PH

k

∣∣∣,s.t. ψk,i

(XkR

12xk

)≤ 0, 1 ≤ i ≤ Ik, 1 ≤ k ≤ K.

(40)



In order to simplify Opt. 3.2, we divide P kC−1x PH

k into

P kC−1x PH

k =

[P 1,1 P 1,2

P 2,1 P 2,2

]. (41)

Combining (39) and (41) leads to

P kC−1x PH

k + P kdiag{{

XHk H

Hk R

−1nkHkXk

}Kk=1

}PH

k

=

[P 1,1 +XH

k HHk R

−1nkHkXk P 1,2

P 2,1 P 2,2 +Ξk

]. (42)

Further exploiting the fundamental properties of matrix deter-minants [27], [56], we have the following equality∣∣∣∣[P 1,1 +XH

k HHk R

−1nkHkXk P 1,2

P 2,1 P 2,2 +Ξk

]∣∣∣∣=∣∣P 2,2 +Ξk

∣∣∣∣XHk H

Hk R

−1nkHkXk +Φk

∣∣, (43)

where

Φk = P 1,1 − P 1,2(P 2,2 +Ξk)−1P 2,1. (44)

Based on (43), the alternating optimization of F k for 1 ≤ k ≤K can be performed. Specifically, the optimization problemOpt. 3.2 is transferred into: for 1 ≤ k ≤ K,

Opt.3.3 : minXk

− log∣∣Φk +XH

k HHk R

−1nk

HkXk

∣∣,s.t. ψk,i

(XkR

12xk

)≤ 0, 1 ≤ i ≤ Ik.

(45)

It can be seen that by exploiting its block diagonal structure,the multiple-matrix-variate matrix-monotonic optimization ofOpt. 3.1 is transferred into several single-matrix-variate matrix-monotonic optimization problems in the form of Opt. 3.3.

For 1 ≤ k ≤ K, by introducing the auxiliary variable

F kQXk= XkR

12xk , (46)

the optimization problem Opt. 3.3 is transferred into:

Opt.3.4: minF k

− log∣∣R− 1

2xk ΦkR

− 12

xk

+QHXk

FHk H

Hk R

−1nk

HkF kQXk

∣∣,s.t. ψk,i (F k) ≤ 0, 1 ≤ i ≤ Ik. (47)

Based on Matrix Inequality 3 in Part I [1], we have

log∣∣R− 1

2xk ΦkR

− 12

xk +QHXk

FHk H

Hk R

−1nk

HkF kQXk

∣∣≤∑j

log|λN−j+1(R− 1

2xk ΦkR

− 12

xk )+λj(FHk H

Hk R

−1nk

HkF k)|

(48)

based on which the optimal unitary matrix QXkequals [17]

Qopt,Xk= USNR,kU

HΦkRk

(49)

where the unitary matrices USNR,k and UHΦkRk

are defined bythe following SVD and EVD,

FHk H

Hk K

−1nkHkF k

= USNR,kΛSNR,kUHSNR,k with ΛSNR,k ↘

R− 1

2xk ΦkR

− 12

xk =UΦkRkΛΦkRk

UHΦkRk

with ΛΦkRk↗ . (50)

From the multi-objective optimization viewpoint, the optimalsolutions of Opt. 3.4 belong to the Pareto optimal solution setsof the following optimization problems for 1 ≤ k ≤ K [17]

Opt.3.5 : maxF k

λ(FH

k HHk R

−1nkHkF k

),

s.t. ψk,i (F k) ≤ 0, 1 ≤ i ≤ Ik.(51)

which is equivalent to the following matrix-monotonic optimiza-tion problem:

Opt.3.6 : maxF k

FHk H

Hk R

−1nkHkF k,

s.t. ψk,i (F k) ≤ 0, 1 ≤ i ≤ Ik.(52)

Based on the fundamental results of the previous sections de-rived for matrix-monotonic optimization, we have the followingresults. Clearly, the optimal Xk equals

Xopt,k = F opt,kQopt,XkR

− 12

xk . (53)


ψk,1 (F k) = F kFHk −Rsk . (54)

Based on Lemma 2 in Section II, we have when the rank ofRsk is not higher than the number of columns and the numberof rows in F k, the optimal solution F opt,k is a square root ofRsk .

2) Joint Power Constraints: We have

ψk,1 (F k) = Tr(F kF

Hk

)− Pk,

ψk,2 (F k) = F kFHk − τkI. (55)

Based on Lemma 2 in Section II, the Pareto optimal solutionsF opt,k satisfy the following structure

F opt,k = V HkΛF k

UHArb,k, (56)

where every diagonal element of the rectangular diagonal matrixΛF k

is smaller than√τk. The diagonal matrix ΛF k

can beefficiently solved using a variant water-filling algorithm [52],[54].

3) Multiple Weighted Power Constraints: We have

ψk,i (F k) = Tr(Ωk,iF kF

Hk

)− Pk,i. (57)

Based on Lemma 3 in Section II, the Pareto optimal solutionsF opt,k satisfy the following structure

F opt,k = Ω− 1

2

k V HkΛF k

UHArb,k, (58)

where Ωk is given by (36), while V Hkis defined by the follow-

ing SVD, respectively,

R− 1

2nk HkΩ

− 12

k =UHkΛHk

VH

Hkwith ΛHk

↘ . (59)

The diagonal matrix ΛF kcan be efficiently solved using water-

filling algorithms [53], [54].Remark 1: The results of this paper can also be applied to

more complex scenarios. For example, when the CSI betweena sensor and its data fusion center is imperfect, Hk = Hk +

HW,kΨ12

k , where Hk and HW,kΨ12

k are the estimated CSI



Fig. 3. Multi-hop cooperative AF MIMO relaying network.

and the channel estimation error, respectively. The correlationmatrix Ψk is a function of both the channel estimator and ofthe training sequence. Based on the proposed framework, theoptimal structures of the optimal solutions for the robust signalcompression matrices at different sensors can also be readilyderived.

V. MULTI-HOP AF MIMO RELAYING NETWORKS

Multi-hop relaying communication is one of the most im-portant enabling technologies for future flexible and high-throughput communications, such as machine-to-machine,device-to-device, vehicle-to-vehicle, internet of things or satel-lite communications [28], [48]. The key idea behind multi-hopcommunications is to deploy multiple relays to realize the com-munications between the source node and destination node [48],[49]. Before presenting our third application of transceiver op-timization for multi-hop communications, we first highlight thedifference between our work presented in this section and theprevious conclusions in [28], [31].� We consider a more general power constraint which in-

cludes both the per-antenna power constraint in [31] andthe shaping constraints in [28] as its special cases.

� The channel estimation errors are realistically taken intoaccount in our work. By contrast, in [31] the CSI is assumedto be perfectly known.

To the best of our knowledge, the robust transceiver optimiza-tion for multi-hop communications even under the per-antennapower constraint is still the problem not yet fully solved in theexisting literature. Therefore, the results presented in this sectionis novel and significant.

TheK-hop AF MIMO relaying network is illustrated in Fig. 3,where the source, denoted as node 0, communicates with thedestination, represented by nodeK, with the help of the (K − 1)relays, which are nodes 1 to (K − 1). Denote the signal sent bythe source asx0, which has the covariance matrix of σ2

x0I . Then

the signal model in thekth hop, for 1 ≤ k ≤ K, can be expressedas

xk = HkXkxk−1 + nk, (60)

where xk is the signal received by node k, Hk is the channelmatrix of the kth hop, and nk is the additive noise of the corre-sponding link with the covariance matrix σ2

nkI , while Xk is the

forwarding matrix of node (k − 1). Note that S1 is the source’s

transmit precoding matrix. When the channel estimation error isconsidered, based on a practical channel estimation scheme [15]the CSI of the kth hop is expressed as

Hk = Hk +HW,kΨ12

k , (61)

where Hk and HW,kΨ12

k are the estimated CSI and the channelestimation error of the kth hop, respectively. Furthermore, Ψk

is the covariance matrix of the channel estimate, and the ele-ments of HW,k follow the independent and identical complexGaussian distribution CN (0, 1). For notational convenience, letus define the new variables F 1QX1

= X1, with the associatedunitary matrix QX1

, and F kQk for 2 ≤ k ≤ K as

F k = XkK12nk−1Mk−1Q

HXk, (62)

where Qk is the associated unitary matrix,

Mk=(K

− 12

nk HkF kFHk H

H

k K− 1

2nk +I

) 12

, (63)

Knk=(σ2nk

+Tr(F kF

Hk Ψk

))I, (64)

and clearly K12n0M0 = σx0

I . Based on these definitions, asproved in Appendix B the MSE matrix of the data detectionat the destination is expressed as, [28], [31]

ΦMSE

({F k}Kk=1, {QXk}Kk=1,C

)= σ2

x0CCH − σ2

x0C

(K∏

k=1

M− 1

2

k K− 1

2nk HkF kQXk

)H

×(

K∏k=1

M− 1

2

k K− 1

2nk HkF kQXk

)CH. (65)

Based on the MSE matrix given in (65), both the linear andnonlinear transceiver optimization problems [28], [31] can beunified into the general optimization problem Opt. 4.1 givenin (66), shown at the bottom of this page. Various objectivefunctions typically adopted for Opt. 4.1 are listed in Table I.For linear transceiver optimization, to realize different lev-els of fairness between different transmitted data streams, ageneral objective function can be formulated as an additivelySchur-convex function [31], [51] or additively Schur-concavefunction [31], [51] of the diagonal elements of the MSE matrix,which are given by Obj. 3 and Obj. 4 [31], [51], respectively. Theadditively Schur-convex function f convexA-Schur (·) and the additivelySchur-concave function f concaveA-Schur (·) represent different levels offairness among the diagonal elements of the data MSE matrix.When nonlinear transceivers are adopted for improving the BERperformance at the cost of increased complexity, e.g., THP orDFE, the objective functions of the transceiver optimization canbe formulated as a multiplicative Schur-convex function or amultiplicative Schur-concave function of the vector consisting

Opt4.1 : min{F k}Kk=1,{QXk

}Kk=1,Cf(ΦMSE


)),

s.t. ψk,i(F k) ≤ 0, 1 ≤ i ≤ Ik, 1 ≤ k ≤ K,[C]i,i = 1, [C]i,j = 0 for i < j, 1 ≤ i ≤ N.

(66)



TABLE ITHE OBJECTIVE FUNCTIONS AND THE ASSOCIATED OPTIMAL FIRST UNITARY MATRICES Q1 FOR MULTI-HOP COOPERATIVE AF RELAY NETWORKS

of the squared diagonal elements of the Cholesky-decompositiontriangular matrix of the MSE matrix, that is, Obj. 5 andObj. 6 [31], [51], respectively, where L is a lower triangularmatrix. The multiplicatively Schur-convex function f convexM-Schur(·)and the multiplicatively Schur-concave function f concaveM-Schur (·) re-flect the different levels of fairness among the different datastreams, i.e., different tradeoffs among the performance of dif-ferent data steams [17]. The detailed definitions of f convexA-Schur (·),f concaveA-Schur (·), f convexM-Schur(·) and f concaveM-Schur (·) are given in Appendix C.This appendix makes our work self-contained.

The constraintsψk,i(F k) ≤ 0 are right unitarily invariant, andthe power constraint model of Opt. 4.1 is more general than thepower constraint models considered in [17], [28], [31].

For linear transceivers with the objective functionsObjs. 1-4 in Table I, C = I is an identity matrix, while fornonlinear transceiver optimization with the objective functionsObj. 5 and Obj. 6 in Table I, C is a lower triangular matrix.Specifically, we assume that the size of C is N ×N . Then, fornonlinear transceivers, the optimal C satisfies [31]

Copt = diag{{[L]i,i}Ni=1

}L−1, (67)

where L is the triangular matrix of the Cholesky decompositionof the following matrix [31]

LLH = ΦMSE

({F k}Kk=1, {QXk}Kk=1

)= σ2

x0I − σ2

x0

(K∏

k=1

M− 1

2

k K− 1

2nk HkF kQXk

)H

×(

K∏k=1

M− 1

2

k K− 1

2nk HkF kQXk

). (68)

The optimal unitary matrices QXkcan be derived based on

majorization theory. Specifically, the optimal Qk for k > 1 arederived as [26], [28], [31]

Qopt,Xk= V Ak

UHAk−1

, (69)

where the unitary matrices V Akand UAk

are defined by thefollowing SVDs

M− 1

2

k K− 1

2nk HkF k = UAk

ΛAkV H

Akwith ΛAk

↘ . (70)

The optimal QX1is determined by the specific objective func-

tion, and various Qopt,X1associated with different objective

functions are also summarized in Table I. Here, the unitarymatrix UArb denotes an arbitrary matrix having the appropriatedimension. The unitary matrixUW is the unitary matrix defined

by the following EVD

W = UWΛWUHW with ΛW ↘ . (71)

The unitary matrix UDFT is a DFT matrix [55], [56]. Finally,the unitary matrix UGMD ensures that the triangular matrix ofthe Cholesky decomposition of ΦMSE({F k}Kk=1, {QXk

}Kk=1)has the same diagonal elements [31].

Given the optimal Qopt,Xkand Copt, the objective function

of Opt. 4.1 can be rewritten as [28]

f(ΦMSE

({F k}Kk=1, {Qopt,Xk}Kk=1,Copt

))= f

⎛⎜⎝⎧⎨⎩

K∏k=1

λi(FHk H

H

k K−1nkHkF k)

1 + λi(FHk H

H

k K−1nkHkF k)

⎫⎬⎭N

i=1

⎞⎟⎠� fEigen

({λ(FH

k HH

k K−1nkHkF k

)}K

k=1

). (72)

In (72) fEigen(·) is a monotonically decreasing function with

respect to the eigenvalue vector λ(FHk H

H

k K−1nkHkF k). The

specific formula of fEigen(·) is determined by the specific per-formance metrics. For example, for sum MSE minimizationfEigen(·) equals

fEigen

({λ(FH

k HH

k K−1nkHkF k

)}K

k=1

)

=

I∑i=1

x20

⎛⎝1−K∏

k=1

λi(FHk H

H

k K−1nkHkF k)

1 + λi(FHk H

H

k K−1nkHkF k)

⎞⎠ . (73)

In addition, for sum rate maximization fEigen(·) equals

fEigen

({λ(FH

k HH

k K−1nkHkF k

)}K

k=1

)

=I∑

i=1

log

⎛⎝1−K∏

k=1

λi(FHk H

H

k K−1nkHkF k)

1 + λi(FHk H

H

k K−1nkHkF k)

⎞⎠ .

(74)

Hence, given Qopt,Xkand Copt, Opt. 4.1 is transferred into

Opt.4.2 : min{F k}Kk=1

fEigen

({λ(FH

k HH

k K−1nkHkF k

)}K

k=1

),

s.t. Knk=(σ2nk

+Tr(F kF

Hk Ψk

))I,

ψk,i(F k) ≤ 0, 1 ≤ i ≤ Ik, 1 ≤ k ≤ K.

(75)



Since the objective function of Opt. 4.2 is a monotonically

decreasing function of λ(FHk H

H

k K−1nkHkF k), it can be decou-

pled into the following sub-problems: for 1 ≤ k ≤ K,

Opt.4.3 : minF k

λ(FH

k HH

k K−1nkHkF k

),

s.t. Knk=(σ2nk+Tr

(F kF

Hk Ψk

))I,

ψk,i(F k) ≤ 0, 1 ≤ i ≤ Ik.

(76)

Clearly, Opt. 4.3 is equivalent to the following matrix-monotonic optimization problem

Opt.4.4 : minF k

FHk H

H

k K−1nkHkF k,

s.t. Knk=(σ2nk+Tr

(F kF

Hk Ψk

))I,

ψk,i(F k) ≤ 0, 1 ≤ i ≤ Ik.

(77)

In this application, by exploiting its cascade structure, we areable to transfer the associated multiple-matrix-variate matrix-monotonic optimization problem into several single-matrix-variate matrix-monotonic optimization problems. Based on thefundamental results of the previous sections, we readily have thefollowing results.


ψk,1(F k) = F kFHk −Rsk . (78)

As proved in Part I, based on Lemma 1 in Section II, it isconcluded that when the rank of Rsk is not higher than thenumber of columns and the number of rows in F k, a suboptimalsolution F opt,k that maximizes a lower bound of the objectiveof Opt. 4.4 is a square root of Rsk . When Ψk = 0 the lowerbound is tight and then the suboptimal solution will be the Paretooptimal solution of Opt. 4.4.

2) Joint Power Constraint: We have

ψk,1 (F k) = Tr(F kF

Hk

)− Pk,

ψk,2 (F k) = F kFHk − τkI.

(79)

As proved in Part I, based on Lemma 2 in in Section II for thegeneral case Ψk �∝ I , a suboptimal solution that maximizes alower bound of the objective of Opt. 4.4 satisfies the followingstructure

F k=σnk

Ψ− 1

2

k V˜Hk

Λ˜F k

UHArb,k(

1−Tr(Ψ

− 12

k ΨΨ− 1

2

k V˜Hk

Λ˜F k

ΛH˜F k

V H˜Hk

)) 12

, (80)

where Ψk = σ2nkI + PkΨk. It is worth noting that when Ψk =

0 or Ψk ∝ I , the corresponding lower bound is tight. In otherwords, in that case the suboptimal solution is exactly the Paretooptimal solution of Opt. 4.4. The unitary matrix V

˜Hkis the

right unitary matrix of the following SVD

Hk

(σ2nkI + PkΨk

)− 12 = U

˜HkΛ

˜HkV H

˜Hk,with Λ

˜Hk↘,

(81)

and every diagonal element of the rectangular diagonal matrixΛ

˜F kin (80) is smaller than the following threshold√τk(σ2nk

+ Pkλmin(Ψk))/(σ2nk

+ Pkλmax(Ψk)). (82)


can be efficiently solved using avariant water-filling algorithm [53], [54].

3) Multiple weighted power constraints: We have

ψk,i (F k) = Tr(Ωk,iF kF

Hk

)− Pk,i. (83)

As proved in Part I, based on Lemma 3 in Section II, we concludethat the Pareto optimal solutions F opt,k satisfy the followingstructure

F opt,k=σnk

Ω− 1

2

k V HkΛ

˜F kUH

Arb,k(1−Tr

(Ω

− 12

k ΨkΩ− 1

2

k V HkΛ

˜F kΛH

˜F kV H

Hk

)) 12

,

(84)

where the unitary matrix V Hkis defined by the SVD

HkΩ− 1

2

k = UHkΛHk

V HHk

with ΛHk↘, (85)

and the matrix Ωk is defined by

Ωk = σ2nk

Ik∑i=1

αk,i (Ωk,i + Pk,iΨk) . (86)


can be efficiently solved using water-filling algorithms [53], [54].

VI. DISCUSSIONS

In this paper, we have investigated three representative ex-amples for the proposed framework of multi-variable matrix-monotonic optimization. Based on the proposed matrix-monotonic framework, the structure of the optimal solutions forthe three largely different optimization problems can be derivedin the same logic. The distinct difference between our workand existing work is that more general power constraints havebeen taken in account. Taking more general power constraintsinto account is definitely not trivial extensions. From physicalmeaning perspective, the considered optimization under moregeneral power constraints includes more MIMO transceiver op-timizations as its special cases. Moreover, from a mathematicalviewpoint, the optimization with more general power constraintsis more challenging. It is impossible to extend the existing resultsin the literature to the conclusions given in this paper via usingsimple substitutions. From convex optimization theory perspec-tive, adding one more constraint may not change the convexityof the considered optimization problem. Specifically, adding onemore linear matrix inequality on a SDP problem, the resultingproblem is still a SDP problem. Adding a quadratical constrainton a QCQP problem, the resulting problem is still a QCQP. Thestory is totally different for the matrix-monotonic optimizationframework as the matrix-monotonic optimization frameworkaims at deriving the structure of the optimal solutions. One moreconstraint will change the feasible region of matrix variate andsignificantly change the structure of the optimal solutions. Thecorresponding analytical derivations will change distinctly.

We also would like to point out that the matrix-monotonicoptimization framework is applicable to more complicated com-munication systems. Recently, in [18] based on the matrix-monotonic optimization framework, a general framework on



hybrid transceiver optimizations under sum power constraintis proposed. Different from the fully digital MIMO systems, ina typical hybrid MIMO system, at the source or the destinationthe precoder or the receiver consists of two parts, i.e., analogpart and digital part. For the analog part, only the phase of thesignal at each antenna is adjustable. After that, in [19] based onthe matrix-monotonic optimization framework, a framework onthe transceiver optimizations for multi-hop AF hybrid MIMOrelaying systems is further proposed. In multi-hop communica-tions, the forwarding matrix at each relay consists of three parts,the left analog part, the inner digital part and the right analogpart.

VII. SIMULATION RESULTS AND DISCUSSIONS

A. Two-User MIMO Uplink

We first consider the MU-MIMO uplink, where a pair of4-antenna mobile users communicate with an 8-antenna BS. Wedefine Pk

σ2n

as the SNR for the kth user, where Pk is the sum

transmit power of userk andσ2n is the noise power at each receive

antenna of the BS. Without loss of generality, the same maximumtransmit power is assumed for all the users, i.e., P1 = P2.Based on the Kronecker correlation model [13]–[15], the spatialcorrelation matrix RRx of the BS’s receive antennas and thespatial correlation matrix RTx,k of the kth user’s transmit an-tennas, where k = 1, 2, are specified respectively by [RRx]i,j =

r|i−j|r and [RTx,k]i,j = r

|i−j|t,k . In the simulations, we further set

rt,1 = rt,2 = rt. Three power constraints, namely, the shap-ing constraint, the joint power constraint and the per-antennapower constraints, are considered. For the shaping constraint, thewidely used Kronecker correlation model of [Rsk ]i,j = 0.6|i−j|

is employed [28]. For the joint power constraint, the threshold ischosen as τk = 1.4. For the per-antenna power constraints, thepower limits for the four antennas of each user are set to 1.2,1.2, 0.8 and 0.8, respectively.

It is worth highlighting that the transceiver optimizationunder these three power constraints can be transferred intoconvex optimization problems, which can be solved numeri-cally using the CVX tool [58]. This approach however suffersfrom high computational complexity, especially for high dimen-sional antenna arrays. By contrast, our approach presented inSection III provides the optimal closed-form solutions for thesame transceiver optimization design problems. Fig. 4 comparesthe sum rate performance as the function of the SNR for theproposed closed-form solutions and for the numerical optimiza-tion solutions computed by the CVX tool. It can be seen thatour closed-form solutions have an identical performance to thesolutions computed by the CVX tool.

B. Signal Compression for Distributed Sensor Networks

In this subsection, we investigate the performance of the pro-posed algorithm employed for signal compression in distributedsensor networks. Specifically, the distributed sensor networkconsidered consists of K sensors and a data fusion center. Eachsensor is equipped with 4 antennas and the data fusion center isequipped with 8 antennas. The per-antenna power constraints

Fig. 4. Sum rate performance comparison between the proposed closed-formsolutions and the solutions computed by the CVX tool for the two-user MIMOUplink.

for the four antennas of each sensor are set to 1.2, 1.2, 0.8and 0.8, respectively. For the signal correlations between dif-ferent sensors, the distance-dependent correlation matrix modelof [27] is adopted. Specifically, we have Rxm,n

= e−dm,nI forthe mth sensor and the for the nth sensor, where dm,n is thecorrelation between these two sensors. In our simulations, dm,n

is distributed uniformly between 0 and 1. In order to quantify theperformance advantages attained, a benchmark algorithm basedon CVX is used in this subsection. The algorithm based on CVXaims for minimizing the weighted sum MSE under per-antennapower constraints, which is termed as the linear minimum meansquare error (LMMSE) algorithm. In the LMMSE algorithm,the signal compression matrices of the different sensors and thecombiner matrix at the data fusion center are optimized itera-tively. At each iteration, the optimization problem consideredis a standard QCQP problem, which can be readily solved byCVX. Observe in Fig. 5 that the proposed algorithm alwaysoutperforms the CVX-based benchmarker.

C. Dual-Hop AF MIMO Relaying Network

A dual-hop AF MIMO relaying network is simulated, whichconsists of one source, one relay and one destination. All thenodes are equipped with 4 antennas. At the source and relay, per-antenna power constraints are imposed. Specifically, the powerlimits for the four antennas are set as 1, 1, 1 and 1, respectively.The SNR in each hop is defined as the ratio between the transmitpower and the noise variance, i.e., SNRk = Pk

σ2nk

. Without loss

of generality, the SNRs in the both hops are assumed to be thesame, namely, SNR1 = SNR2 = SNR.

In contrast to the existing works [28], [31], which considerthe transceiver optimization unrealistically with the perfect CSI,in this paper, we focus on the robust transceiver optimiza-tion, which takes into account the channel estimation error.In the simulations, the estimated channel matrix is generated

according to Hk = HW,kΨ12

k [17], where we have [Ψk]i,j =



Fig. 5. Mutual information performance comparisons between the proposedalgorithm and the LMMSE algorithm based on CVX for distributed sensornetworks with different numbers of sensors.

Fig. 6. Sum rate performance comparison between our proposed robust designand the non-robust design of [31] for the dual-hop AF MIMO relaying network.

0.6|i−j|. The elements of HW,k are independently identicallydistributed Gaussian random variables. In order to ensure thatE{[H]i,j [H]∗i,j} = 1, ∀i, j, we set E{[HW,k]i,j [HW,k]

∗i,j} =

σ2ek

and E{[HW,k]i,j [HW,k]∗i,j} = 1− σ2

ek. Without loss of

generality, we assumeσ2e1

= σ2e2

= σ2e . It can be seen from Fig. 6

that our robust design achieves better sum rate performancethan the non-robust design of [31]. Furthermore, as expected,the performance gap between the robust and non-robust designsbecomes larger as the channel estimation error increases.

VIII. CONCLUSION

In this paper, we investigated the application of the frameworkof matrix-monotonic optimization in the optimizations with

multiple matrix-variates. It is shown that when several propertiesare satisfied, the framework of matrix-monotonic optimizationstill works, based on which the optimal structures of multi-ple matrix-variates can be derived. Then the multiple matrix-variable optimizations can be effectively solved in iterative man-ners. Three specific examples are also given in this paper to ver-ify the validity of the proposed multi-variable matrix-monotonicoptimization framework. Specifically, under various power con-straints, i.e., sum power constraint, shaping constraints, jointpower constraints and multiple weighted power constraints, thetransceiver optimizations for uplink MIMO communications,the compression matrix optimizations for distributed sensornetworks, and the robust transceiver optimizations for multi-hopAF MIMO relaying systems have been investigated. At the endof this paper, several numerical results demonstrated the accu-racy and performance advantages of the proposed multi-variablematrix-monotonic optimization framework.

APPENDIX ACOMPUTATION OF P k AND Ξk

Given the following block diagonal matrix

Φ = diag{{

XHk H

Hk R

−1nkHkXk

}Kk=1

}(87)

the permutation matrix P k aims at changing the orders ofthe kth element XH

k HHk R

−1nkHkXk and the first element

XH1 H

H1 R

−1n1H1X1 along the diagonal line. Before construct-

ing P , we first give an identity matrix I that has the samedimensions as Φ. Moreover, I can be interpreted as a blockdiagonal matrix as

I = diag{{Ik}Kk=1

}(88)

where Ik is an identity matrix of the same dimensions asXH

k HHk R

−1nkHkXk for 1 ≤ k ≤ K. Moreover, I is further

divided into the following submatrices

I = diag{{Ik}Kk=1

}=

⎡⎢⎢⎢⎣I1

I2

...IK

⎤⎥⎥⎥⎦ (89)

where Ik and Ik have the same row number for 1 ≤ k ≤ K.Based on the above definitions of Ik’s, we have

IkΦIHj = IkΦIT

j = 0, for, k �= j, (90)

and

IkΦIHk = IkΦIT

k = XHk H

Hk R

−1nkHkXk. (91)



Therefore, based on (89) P k is constructed by interchanging I1

and Ik, i.e.,

P k =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ik

I2

...Ik−1

I1

Ik+1

...IK

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. (92)

It is obvious that P k is a unitary matrix, i.e.,

P kPHk = I and PH

k P k = I. (93)

Based on (92) and together with (90) and (91), we have

P kdiag{{

XHk H

Hk R

−1nkHkXk

}Kk=1

}PH

k

=

[XH

k HHk R

−1nkHkXk 0

0 Ξk

]. (94)

where Ξk is the following block diagonal matrix

Ξk = diag{Φ2, . . . , Φk−1, Φ1, Φk+1, . . . , ΦK} (95)

with Φj = XHj H

Hj R

−1njHjXj .

APPENDIX BMSE MATRIX FOR MULTI-HOP COMMUNICATIONS

Based on the signal model given in (60), at the destination thereceived signal y equals

y = xK = HKXKxK−1 + nK . (96)

After performing a linear equalizer G, the signal estimationMSE matrix at the destination can be written in the followingformula [26], [28], [31]

ΦMSE

(G, {Xk}Kk=1,C

)= E{(Gy −Cx0)(Gy −Cx0)

H} (97)

where C = I +B and B is a strictly lower triangular ma-trix [17]. For the linear transceivers, B is a constant matrix,i.e., B = 0. On the other hand for the nonlinear transceiverswith THP or DFE, B corresponds to the feedback operationsand should be optimized as well [26], [28], [31]. Substituting(96) into ΦMSE(G, {Xk}Kk=1) in (97), we have

ΦMSE

(G, {Xk}Kk=1,C

)= G

(HKXKRxK−1

XHKH

H

K

+Tr(XKRxK−1XH

KΨK)I

)GH

−G

(K∏

k=1

HkXk

)Rx0

CH−CRx0

(K∏

k=1

HkXk

)H

GH

+GRnKGH +CRx0

CH, (98)

where Rxk= E{xkx

Hk }. The corresponding LMMSE equal-

izer GLMMSE equals

GLMMSE = CRx0

(K∏

k=1

HkXk

)H

×(HKXKRxK−1

XHKH

H

K +KnK

)−1

(99)

with

KnK= Tr(XKRxK−1

XHKΨK)I +RnK

. (100)

It is well-known that the LMMSE equalizer GLMMSE is theoptimal G for ΦMSE(G, {Xk}Kk=1) as [17]

ΦMSE

(G, {Xk}Kk=1,C

) � ΦMSE

(GLMMSE, {Xk}Kk=1,C

).

(101)

Substituting GLMMSE into (98), we have

ΦMSE

(G, {Xk}Kk=1,C

)= CRx0

CH −CRx0

(K∏

k=1

HkXk

)H

×(HKXKRxK−1

XHKH

H

K +KnK

)−1

×(

K∏k=1

HkXk

)Rx0

CH. (102)

Therefore, based on the definition ofF k in (62) and the definitionof Mk in (63) we have

ΦMSE

(G, {Xk}Kk=1,C

)= ΦMSE


)= σ2

x0CCH − σ2

x0C

(K∏

k=1

M− 1

2

k K− 1

2nk HkF kQXk

)H

×(

K∏k=1

M− 1

2

k K− 1

2nk HkF kQXk

)CH. (103)

APPENDIX CFUNDAMENTAL DEFINITIONS OF MAJORIZATION THEORY

A brief introduction of majorization theory is given in this ap-pendix. Generally speaking, majorization theory is an importantbranch of matrix inequality theory [57]. Majorization theory isa very useful mathematical tool to prove the inequalities for thediagonal elements of matrices, the eigenvalues of matrices andthe singular values of matrices. Majorization theory can revealthe relationships between diagonal elements and eigenvalues,based on which some extrema can be computed. Moreover,majorization theory can quantitatively analyze the relationshipsbetween the eigenvalues or singular values of matrix productsand matrix additions and that of the involved individual matrices.Based on majorization theory, a rich body of useful matrixinequalities can be derived, based on which the extrema of the



matrix variate functions can be derived. The definitions of addi-tively Schur-convex, additively Schur-concave, multiplicativelySchur-convex and multiplciatively Schur-concave functions aregiven in the following. Meanwhile, we would like to point outthat Schur-convex function is a kind of increasing function andSchur-concave function is a kind of decreasing function [28].They actually have no relationship with the traditional convexor concave properties defined in the convex optimization the-ory [50].

Definition 1 ([57]): For a K × 1 vector x ∈ RK , the �th

largest element of x is denoted as x[�], i.e., x[1] ≥ x[2] ≥ · · · ≥x[K]. Based on this definition, for twoK × 1vectorsx,y ∈ R

K ,the statement that y majorizesx additively, denoted byx ≺+ y,is defined as follows

m∑n=1

x[n] ≤m∑

n=1

y[n], m = 1, . . . ,K−1, andK∑

n=1

x[n] =K∑

n=1

y[n].

(104)

Definition 2 ([57]): A real function f(·) is additively Schur-convex when the following relationship holds

f(x) ≤ f(y) when x ≺+ y. (105)

A real function f(·) is additively Schur-concave if and only if−f(·) is additively Schur-convex.

Definition 3 ([28], [51]): Given K × 1 vectors x,y ∈ RK

with nonnegative elements, the statement that the vector y ma-jorizes vector x multiplicatively, denoted by x ≺× y, is definedas follows

m∏n=1

x[n] ≤m∏

n=1

y[n], m=1, . . . ,K−1, andK∏

n=1

x[n]=

K∏n=1

y[n].

(106)

Definition 4 ([28], [51]): A real function f(·) is multiplica-tively Schur-convex when the following relationship holds

f(x) ≤ f(y) when x ≺× y. (107)

A real function f(·) is multiplicatively Schur-concave if andonly if −f(·) is multiplicatively Schur-convex.

Generally, it is not convenient to use these definitions to provewhether a function is Schur-convex or not. In the following,two criteria are given, based on which we can judge whether afunction is additively Schur-convex or multiplicatively Schur-convex [17], [25], [26]. For a given function f(·), according tothe value order of the elements of x the considered functionf(x) is first reformulated as

f(x) = ψ(x[1], . . . , x[k], x[k+1], · · · ). (108)

When f(x) = ψ(x[1], . . . , x[k] − e, x[k+1] + e, · · · ) is a de-creasing function with respect to e for e ≥ 0 and x[k] − e ≥x[k+1] + e, f(·) is additively Schur-convex. On the other hand,when f(x) = ψ(x[1], . . . , x[k]/e, x[k+1]e, · · · ) is a decreasingfunction with respect to e for e ≥ 1 and x[k]/e ≥ x[k+1]e, f(·)is multiplicatively Schur-convex.

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Chengwen Xing (Member, IEEE) received theB.Eng. degree from Xidian University, Xi’an, China,in 2005, and the Ph.D. degree from The University ofHong Kong, Hong Kong, in 2010.

Since September 2010, he has been with the Schoolof Information and Electronics, Beijing Institute ofTechnology, Beijing, China, where he is currently aFull Professor. From September 2012 to December2012, he was a Visiting Scholar with the Universityof Macau. His current research interests include sta-tistical signal processing, convex optimization, mul-

tivariate statistics, combinatorial optimization, massive MIMO systems, andhigh-frequency band communication systems. He is an Associate Editor for theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, KSII Transactions on In-ternet and Information Systems, Transactions on Emerging TelecommunicationsTechnologies, and China Communications.

Shuai Wang (Member, IEEE) received the B.Eng.(Hons.) degree in communications engineering fromZhengzhou University, Zhengzhou, China, in 2005,and the Ph.D. (Hons.) degree in communicationsengineering from the Beijing Institute of Technology,Beijing, China, in 2012. From 2010 to 2011, he was aVisiting Ph.D. Student with the School of Electronicsand Computer Science, University of Southampton,Southampton, U.K. He has been with the School of In-formation Science and Electronics, Beijing Instituteof Technology, since July 2012, where he currently

holds a post of Associate Professor. He has authored/coauthored more than20 peer-reviewed articles, mainly in leading IEEE journals or conferences.He also holds 17 patents. His research interests include channel estimation,anti-jamming transmission, synchronization techniques, and beamforming. Heserved or is serving as the Principal Investigator for 12 research funds, includingtwo granted by the National Science Foundation of China. He was a recipientof the (Second Class) Scientific and Technical Progress Award granted by theMinistry of Industry and Information Technology of China. He was a recipientof the award for outstanding Ph.D. dissertation granted by the Beijing MunicipalEducation Commission in 2013 with other 49 co-winners, nominated from allthe Ph.D. graduates that received their degree in Beijing.



Sheng Chen (Fellow, IEEE) received the B.Eng.degree in control engineering from the East ChinaPetroleum Institute, Dongying, China, in 1982, andthe Ph.D. degree in control engineering from theCity, University of London, London, U.K., in 1986.In 2005, he was awarded the higher doctoral de-gree, Doctor of Sciences (D.Sc.), from the Universityof Southampton, Southampton, U.K. From 1986 to1999, he held research and academic appointmentswith the Universities of Sheffield, Edinburgh andPortsmouth, all in U.K. Since 1999, he has been with

the School of Electronics and Computer Science, University of Southamp-ton, where he holds the post of a Professor in intelligent systems and signalprocessing. He has authored/coauthored more than 700 research papers. Hisresearch interests include neural network and machine learning, adaptive signalprocessing, wireless communications, and nonlinear system modelling. He isa Fellow of the United Kingdom Royal Academy of Engineering, a Fellow ofIET, a Distinguished Adjunct Professor at King Abdulaziz University, Jeddah,Saudi Arabia, and an original ISI highly cited Researcher in engineering (March2004). He has 15 300+ Web of Science citations with h-index 54, and 30 900+Google Scholar citations with h-index 75.

Shaodan Ma (Member, IEEE) received the doublebachelor’s degrees in science and economics and themaster’s degree of Engineering from Nankai Univer-sity, Tianjin, China, in 1999 and 2002, respectively,and the Ph.D. degree in electrical and electronic en-gineering from The University of Hong Kong, HongKong, in 2006. From 2006 to 2011, she was a Post-doctoral Fellow with The University of Hong Kong.She was a Visiting Scholar with Princeton Universityin 2010. Since August 2011, she has been with theUniversity of Macau, Macao, where she is currently

an Associate Professor. Her research interests include signal processing andcommunications, particularly, transceiver design, resource allocation, and per-formance analysis.

H. Vincent Poor (Fellow, IEEE) received the Ph.D.degree in electrical engineering and computer sciencefrom Princeton University, Princeton, NJ, USA, in1977. From 1977 to 1990, he was on the facultyof the University of Illinois at Urbana–Champaign.Since 1990, he has been on the faculty of PrincetonUniversity, where he is currently the Michael HenryStrater University Professor of electrical engineering.From 2006 to 2016, he served as the Dean of Prince-ton’s School of Engineering and Applied Science. Hehas also held visiting appointments with several other

institutions, including most recently at Berkeley and Cambridge. His researchinterests include information theory, machine learning and network science,and their applications in wireless networks, energy systems, and related fields.Among his publications in these areas is the forthcoming book Advanced DataAnalytics for Power Systems (Cambridge University Press, 2021).

Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, and is a foreign member of the Chinese Academyof Sciences, the Royal Society, and other national and international academies.He was the recipient of the Technical Achievement and Society Awards ofthe IEEE Signal Processing Society in 2007 and 2011, respectively. Recentrecognition of his work includes the 2017 IEEE Alexander Graham Bell Medal,the 2019 ASEE Benjamin Garver Lamme Award, a D.Sc. honoris causa fromSyracuse University, awarded in 2017, and a D.Eng. honoris causa from theUniversity of Waterloo, awarded in 2019.

Lajos Hanzo (Fellow, IEEE) received the master’sand Doctorate degrees from the Technical University(TU) of Budapest, Budapest, Hungary, in 1976 and1983, respectively. He was also awarded the Doc-tor of Sciences (D.Sc.) degree by the University ofSouthampton, Southampton, U.K., in 2004, and Hon-orary Doctorates by the TU of Budapest in 2009 andby The University of Edinburgh, Edinburgh, U.K., in2015. He has authored/coauthored 1900+ contribu-tions at IEEE Xplore, 19 Wiley-IEEE Press books,and has helped fast-tracking the career of 123 Ph.D.

students. Over 40 of them are Professors at various stages of their careers inacademia and many of them are leading Scientists in the wireless industry.He is Fellow of the Royal Academy of Engineering, IET, and EURASIP. Heis a Foreign Member of the Hungarian Academy of Sciences and a formerEditor-in-Chief of the IEEE Press. He has served several terms as Governor ofboth IEEE ComSoc and of VTS.


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