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Robust beamforming designs for multiuser MISO downlink with perantenna power constraints
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 204 (2015)
Abstract
This paper studies the robust beamforming designs for a multiuser multipleinput singleoutput (MISO) downlink system. Different from the conventional sumpower constraint across all transmit antennas, we consider individual power constraints per antenna at the base station. Assuming that the channel uncertainty is bounded by a spherical region, we develop the optimal robust designs to maximize the minimum worstcase signaltointerferenceplusnoise ratio (SINR) among all users. Specifically, we show that the optimal maxmin SINR beamformers can be obtained by solving a sequence of “dual” minmax power problems. Relying on the Sprocedure and the linear matrix inequality representation for the cone of Lorentzpositive maps (LPMs), respectively, two designs, which are referred to as RobustSP and RobustLPM, are proposed to efficiently solve such minmax power problems. Building on either RobustSP or Robust LPM, a bisection search algorithm is then developed to find the robust maxmin SINR beamformers with guaranteed global optimality and geometrically fast convergence speed. Using the maxmin SINR solutions as a cornerstone, we further put forth the optimal robust design for the worstcase weighted sumrate (WSR) maximization. By formulating the worstcase WSR maximization problem into a monotonic program (MP), we develop a polyblock outer approximation algorithm to obtain the globally optimal solution. Numerical results are presented to demonstrate the merits of the proposed robust beamforming designs.
1 Introduction
For a multiuser wireless downlink with multiple antennas at the base station (BS), transmit beamforming is a lowcomplexity solution to provide spatial diversity and to reduce cochannel interference [1]. Under the assumption of perfect channel state information (CSI), the optimal and suboptimal beamforming designs were developed to maximize the minimum user signaltointerferenceplusnoise ratio (SINR) [1–4] or to maximize the weighted sum of user rates [5–7]. However, perfect CSI is usually unavailable in practical systems due to many practical factors, such as inaccurate channel estimation, quantization of CSI, erroneous or outdated feedback, and time delays or frequency offsets between the reciprocal channels. For these reasons, robust linear and nonlinear beamforming (precoding) designs to combat against channel uncertainty have received intensive research interests for different multiuser multipleinput multipleoutput (MIMO) (single/multicell downlink or relay) systems [8–19]. Generally, robust beamforming is addressed by either a stochastic or a worstcase approach. In the stochastic approach, the CSI uncertainty is often modeled as a random variable subject to a known probability distribution, and the robustness can be provided based on optimizing the average or the outage performance such as sum meansquareerror (MSE) [8] and weightedsumrate (WSR) [9, 10]. On the other hand, the worstcase approach assumes that the CSI lies in a bounded uncertainty region. In this case, the robustness is achieved by optimizing the system under the worstcase channel condition and it usually leads to a minmax or maxmin problem formulation. With this approach, robust linear beamforming designs were developed for maxmin SINR [11–16] or WSR maximization [17–19].
All the existing beamforming designs in [1–19] were developed under a sumpower constraint across all transmit antennas. However, in physical implementation of a multiantenna BS, each antenna usually has its own power amplifier in the analog frontend, and it is individually limited by the linearity of this amplifier [20]. Hence, instead of the sumpower constraint, this leads to power constraints imposed on a perantenna basis. Under such more realistic power constraints, the transmitter optimization was addressed in [20], where the elegant uplinkdownlink duality under a sumpower constraint was extended to downlink problems with perantenna power constraints. The robust sumMSE minimizations with per BS antenna and per BS power constraints were investigated in [21] through the downlinkuplink duality. By establishing the MSE downlinkinterference duality, the authors in [22] extended the work of [21] to solve the weighted sumMSE minimization and minmax MSE problems under general power constraints for multiuser MIMO systems. Downlink beamforming designs under perantenna power constraints were also addressed in some other works. MoorePenrose zero forcing (ZF)based beamforming design was derived to maximize the minimum user rate in [23]. Aiming to minimize the stochastic or worstcase sumMSE, a nonlinear TomlinsonHarashima precoding (THP) design was investigated in [24] under perantenna power constraints. Based on block coordinate ascent and signomial programming methods, beamforming designs were put forth to maximize the WSR for a multiuser downlink with power constraints per antenna groups in [25]. For a large antenna array at the BS, an achievable rate for singleuser multipleinput singleoutput (MISO) beamforming under perantenna constantenvelop constraints was derived in [26]. With perantenna array power constraints, the surrogate duality of the maxmin beamforming was investigated in [27].
In this paper, we investigate the robust transmit beamforming designs for a MISO downlink with normbounded uncertain CSI at the BS. Under the perantenna power constraints, the robust designs aim to maximize the minimum worstcase SINR among all users, or to maximize the worstcase WSR. The main contributions are summarized as follows:

This paper shows that the robust maxmin SINR beamformers can be obtained by solving a sequence of “dual” robust minmax power problems. Building on the solutions to these “dual” problems, an efficient bisection search algorithm can thus be applied to find the optimal maxmin SINR solution.

Two robust designs (i.e., RobustSP and RobustLPM) are developed to solve the “dual” robust minmax power problem. This problem is wellknown nonconvex due to the infinitely many SINR constraints. For RobustSP, we use the Sprocedure to convert the problem into a rankconstrained semidefinite program (SDP), and then apply the SDP relaxation technique to find its (near)optimal solution. Like [15], we give a computable CSI uncertainty bound which ensures the tightness of the SDP relaxation. For RobustLPM, we consider a slightly conservative problem reformulation. Relying on a linear matrix inequality (LMI) representation for the cone of Lorentzpositive maps (LPMs), the new problem is shown to be equivalently transformed into a convex SDP which can be efficiently solved with guaranteed global optimality.

We further put forth the robust beamforming for the worstcase WSR maximization and formulate this problem into a monotonic program (MP). Relying on solving a sequence of robust maxmin SINR problems, a polyblock outer approximation algorithm is developed to find the globally optimal solution.
The remainder of this paper is organized as follows. Section 2 presents the system model. Section 3 introduces the proposed approach to the optimal maxmin SINR beamforming design with perfect CSI. Section 4 gives the problem formulation of the optimal robust maxmin SINR beamforming, and then develops RobustSP and RobustLPM to solve its “dual” minmax power problem. The robust design for the worstcase WSR maximization is presented in Section 5. Numerical results are provided in Section 6, followed by the conclusion in Section 7.
Notations: Uppercase and lowercase boldface letters denote matrices and vectors, respectively. All vectors are column vectors. \(\mathbb {R}^{n}\) and \(\mathbb {C}^{n}\) are the ndimensional spaces of real and complex vectors, respectively. \(\mathbb {E}[{\cdot }]\) denotes the expectation, (·)^{T} is the transpose, (·)^{H} is the Hermitian transpose, · is the absolute value, and ∥·∥ is the Euclidean norm of a vector. 0 and 1 are allzero and allone vectors of appropriate dimension, respectively. \(\mathcal {R}(\cdot)\) and \(\mathcal {I}(\cdot)\) denote the real part and imaginary part of the argument, respectively. The letter j represents the imaginary unit (i.e., \(j=\sqrt {1}\)), while the letter i often serves as an index in this paper. I _{ n } denotes an n×n identity matrix. A≽0 means that A is Hermitian and positive semidefinite, [A]_{ n,n } is the (n,n)th entry of A. tr(A) and rank(A) are the trace and rank of A, respectively. ⊗ denotes the Kronecker product. A circular symmetric complex Gaussian (CSCG) random vector a with mean \(\boldsymbol {\bar {a}}\) and covariance matrix Σ is denoted as \(\boldsymbol {a} \sim \mathcal {CN}(\boldsymbol {\bar {a}}, \boldsymbol {\Sigma })\). ∪ and ∩ denote the union and intersection of set and set, respectively. ∖ denotes the part of that does not intersect with. ⊂ denotes that is contained in. Finally, the elementwise (strictly) inequality for x, y is denoted as x≤y (x<y).
2 Modeling preliminaries
Consider a wireless multiuser MISO downlink system where the BS equipped with M antennas simultaneously transmits K independent signals to K singleantenna users, as shown in Fig. 1. The baseband equivalent representation of the received signal at user k is expressed as:
where \(\boldsymbol {h}_{k} \in \mathbb {C}^{M}\) is the channel vector from the BS to user k, \(\boldsymbol {x}\in \mathbb {C}^{M}\) is the transmitted signal at the BS, \(n_{k}\sim \mathcal {CN}(0, {\sigma _{k}^{2}})\) is the additive white complex Gaussian noise. Assuming that a linear beamforming strategy is employed at the BS, the transmitted signal x at the BS is then given by
where \(\boldsymbol {w}_{k}\in \mathbb {C}^{M}\) and \(s_{k} \in \mathbb {C}\) are the beamforming vector and the information signal for user k, respectively. Without loss of generality (w.l.o.g.), we assume that s _{ k }, ∀k, has a unit power, i.e., \(\mathbb {E}[s_{k}^{2}] = 1\).
Suppose that each transmit antenna has its own power constraint P _{ m }, ∀m. Given the beamformers w _{ k }, ∀k, such perantenna power constraints can be expressed as
Define W:=[w _{1},·,w _{ K }], then the received SINR at user k in terms of W is given by
3 Maxmin SINR beamforming with perfect CSI
As a good starting point, we first develop an efficient approach to find the optimal beamformers that maximize the minimum SINR among all users under the perantenna power constraints (3), when the perfect CSI is available at the BS.
Based on the SINRs in (4), we consider the following classic maxmin SINR balancing problem:
where γ _{ k }>0 is the prescribed SINR target for user k. Define the M×M square matrix D _{ m } with
The perantenna power constraints of (5) are then rewritten as
which are convex quadratic inequality constraints with respect to w _{ k }.
Due to the nonconvexity of SINR_{ k }(W), the maxmin SINR problem (5) is nonconvex, and hence difficult to solve. Yet, we next show that it can be solved by alternatively solving the tractable “dual” perantenna power balancing problems.
For a given scalar λ>0, as shown in [28], by introducing an auxiliary variable α, we formulate the following perantenna power balancing problem:
where the SINR constraints in (8) can be rewritten as:
Observe that an arbitrary phase rotation can be added to the beamformers without affecting the SINRs. Hence, we choose phases such that \(\boldsymbol {h}_{k}^{H} \boldsymbol {w}_{k}, \forall k\), are real and nonnegative. By taking the square root operation for both sides of (9), the constraints then become convex secondorder cone (SOC) (as known as Lorentz cone) constraints [29]. As a result, (8) can be reformulated into the following convex SOCP problem:
which can be efficiently solved by interiorpoint methods with given parameter λ [29]. Note that, with the given γ _{ k }, ∀k, the problem (10) may be infeasible for large positive λ, since that the SINR constraints cannot be satisfied. In this case, we define the optimal value α ^{∗}(λ):=+∞. When (10) is feasible, we define W ^{∗}(λ) as the optimal solution to (10) for a given λ. It can be shown that the function α ^{∗}(λ) obeys the following two properties^{1}:

Monotonicity: If 0<λ _{1}≤λ _{2}, then α ^{∗}(λ _{1})≤α ^{∗}(λ _{2});

Optimum condition: If it holds α ^{∗}(λ)=1, then λ and the corresponding W ^{∗}(λ) for (10) are the optimal value and the optimal solution for (5), respectively.
The two properties clearly indicate that the optimal solution of (5) can be obtained by solving the equation α ^{∗}(λ)=1. Building on the solution for (10), an efficient bisection search algorithm can be then implemented to find the maxmin SINR beamformers for (5) with guaranteed global optimality and geometrically fast convergence speed [28].
It is worth mentioning that the maxmin SINR problem (5) can be equivalently formulated as a quasiconcave problem:
where the phases of w _{ k },∀k, are chosen such that \(\boldsymbol {h}_{k}^{H}\boldsymbol {w}_{k},\forall k\), are real and nonnegative. Clearly, (11) can be efficiently solved using a similar bisection search, where each step involves checking a convex feasibility problem with an updated λ:
We note that (10) and (12) correspond to a similar SOCP; hence, a similar computational effort is required to solve each instance of (10) and that of (12), practically using a software like CVX [29]. Also since that the similar bisectional search methods are applied in the proposed scheme and the feasibility check approach, it is expected that both algorithms can achieve the optimal solution of (5) after the same iterations. While the proposed scheme can solve the maxmin SINR problem with similar computational complexity compared to the feasibility check approach, it provides a new and important insight for the close relationship between the maxmin SINR problem and the minmax power problem in beamforming designs. This allows us to freely choose a more tractable form as the corner stone to pursue the optimal beamforming designs under various important criteria such as rate maximization, MSE minimization, and biterrorrate (BER) minimization, as detailed in Section 5.
4 Robust maxmin SINR beamforming design
In this section, we generalize the approach in the perfect CSI case to the robust downlink beamforming designs where the BS has the bounded CSI uncertainty. We firstly formulate a robust maxmin SINR beamforming problem and show that the optimal solution can be obtained by solving a sequence of its “dual” minmax power problems. An efficient bisection search algorithm is thus developed. Resorting to the Sprocedure and the LMI representation for the cone of LPMs, respectively, RobustSP and RobustLPM are then proposed to solve these robust minmax power problems.
4.1 Robust maxmin SINR beamforming problem
The downlink CSI uncertainty at the BS could be caused by estimation errors, feedback quantization, hardware deficiencies, and delays in CSI acquisition [11]. Motivated by these considerations, we assume the following additive error model:
where \(\boldsymbol {\hat {h}}_{k}\) is the estimated channel at the BS, and δ _{ k } denotes the channel uncertainty. Similar to [11, 13, 15], we further assume that δ _{ k } is bounded by a spherical region^{2}:
where the real parameter ε _{ k }>0 specifies the radius of \(\mathcal {H}_{k}\).
Next, based on the CSI uncertainty region \(\mathcal {H}_{k}\), define the worstcase SINR for user k in terms of W as:
We consider the robust designs aiming to maximize the minimum worstcase SINR among all users. By replacing the SINRs in (5) with the worstcase SINRs in (15), we have the following robust maxmin SINR balancing formulation:
Again the problem (16) is nonconvex. It is worth mentioning that, like the perfect CSI case, (16) can be converted into a quasiconcave problem with the aid of the Sprocedure and rankrelaxation, and the (near)optimal solution can be found by iteratively checking the feasibility of an SDP. Here, we show that (16) can be solved by alternatively solving a sequence of the minmax power problems. For a given λ>0, we consider:
where P _{ m } serves as a “power target” for antenna m and \(\tilde {\alpha }^{*}(\lambda)\) denotes the optimal value of problem (17). Again the problem (17) may be infeasible for large nonnegative λ. In this case, we define \(\tilde {\alpha }^{*}(\lambda):=+\infty \). As with the perfect CSI case, we can establish the following property:
Lemma 1.
\(\tilde {\alpha }^{*}(\lambda)\) is a strictly increasing function for λ>0.
Proof.
Please see Appendix 1.
Let W ^{∗}(λ) denote the optimal beamforming matrix for (17) with the given λ>0. Relying on the monotonicity of \(\tilde {\alpha }^{*}(\lambda)\), we can further show the following close relationship between (16) and (17): □
Lemma 2.
If it holds \(\tilde {\alpha }^{*}({\lambda })=1\), then λ and the corresponding \(\boldsymbol {W}^{*}({\lambda })=[\boldsymbol {w}^{*}_{1}({\lambda }),\cdot,\boldsymbol {w}^{*}_{K}({\lambda })]\) are the optimal value and the optimal solution for (16), respectively.
Proof.
Please see Appendix 2. □
Similar to the perfect CSI case, Lemma 2 indicates that the optimal solution to (16) can be obtained by solving the equation \(\tilde {\alpha }^{*}({\lambda })=1\). Therefore, a bisection search over λ can be then implemented to obtain the robust maxmin SINR beamformers. We describe it in Algorithm 1 in detail.
For Algorithm 1, an appropriate interval of λ ^{∗} should be determined. Obviously, λ _{min}=0, a lower bound of the interval. An upper bound λ _{max} is determined as follows. It is clear that,
This implies that
With such λ _{min} and λ _{max}, the bisection search requires \(\mathcal {O}(\log _{2}((\lambda _{\max }\lambda _{\min })/\delta))\) iterations (e.g., see [29] page 146) to solve \(\tilde {\alpha }^{*}(\lambda)=1\) up to a desired accuracy level, due to the monotonicity of \(\tilde {\alpha }^{*}(\lambda)\) with respect to λ>0. Clearly, Algorithm 1 converges to the optimal solution geometrically fast.
Summarizing, we have the following proposition [29]:
Proposition 1.
Algorithm 1 converges geometrically fast to the global optimal solution W ^{∗} for (16).
To implement Algorithm 1, it is desirable to efficiently solve problem (17) with any given λ>0. Introducing an auxiliary variable α, we rewrite (17) as:
Although (19) is still nonconvex due to the infinitely many SINR constraints, two efficient designs will be provided in the next two subsections.
4.2 The RobustSP design
The basic idea of RobustSP is to firstly convert (19) into a rankconstrained SDP by applying the wellknown Sprocedure [29, 30], and then the converted problem is (near)optimally solved via the SDP relaxation [31].
Based on the definitions of \(\mathcal {H}_{k}\) and \(\widetilde {\text {SINR}}_{k}\), the constraints \({\widetilde {\text {SINR}}_{k}(\boldsymbol {W})}/{\gamma _{k}} \geq \lambda \) can be rewritten as:
where
Next, we will show that the infinitely many constraints (20) have an equivalent LMI representation. To this end, we resort to the wellknown Sprocedure in optimization. This procedure provides an LMI representation for a robust quadratic constraint over an uncertainty set defined by one or two quadratic inequalities. For convenience, we cite the Sprocedure in [30] and [29] below:
Lemma 3.
(Sprocedure) Let A and B be two n×n Hermitian matrices, \(\boldsymbol {c}\in \mathbb {C}^{n}\) and \(d\in \mathbb {R}\). Then the following two conditions are equivalent:

x ^{H} A x+c ^{H} x+x ^{H} c+d≥0 for all \(\boldsymbol {x}\in \mathbb {C}^{n}\) such that x ^{H} B x≤1;

There exists a \(t \in \mathbb {R}\) such that
$$t \geq 0, \quad\left[ \begin{array}{cc} \boldsymbol{A}+t\boldsymbol{B} & \boldsymbol{c} \\ \boldsymbol{c}^{H} & dt \end{array}\right] \succeq \boldsymbol{0}. $$
By applying Lemma 3, problem (19) can be reformulated as an SDP with rank constraints. Define \(\boldsymbol {X}_{k}:=\boldsymbol {w}_{k} \boldsymbol {w}_{k}^{H}\), ∀k. Then X _{ k }≽0 and rank(X _{ k })=1. Using the above Sprocedure, (20) can be transformed as Γ _{ k }(λ)≽0 with respect to λ:
where t _{ k }≥0, ∀k, and
Using the property that \(\boldsymbol {w}_{k}^{H}\boldsymbol {D}_{m}\boldsymbol {w}_{k}=\text {tr}(\boldsymbol {D}_{m}\boldsymbol {X}_{k})\), we get the following equivalent SDP linear constraints:
Until now, we are ready to show the following proposition:
Proposition 2.
(RobustSP) The problem (19) can be equivalently reformulated as a rankconstrained SDP:
From Proposition 2, it is evident that, by dropping the rankone constraints, RobustSP (23) becomes an SDP which can be solved in a numerically reliable and efficient fashion [29]. Suppose \(\boldsymbol {X}_{k}^{*},\forall k\), are the optimal solutions to the SDP relaxation of (23). If \(\boldsymbol {X}_{k}^{*}=\boldsymbol {w}_{k}^{*}\boldsymbol {w}_{k}^{*H}, \forall k\), i.e., the SDP relaxation is tight, then we obtain the optimal \(\boldsymbol {w}_{k}^{*},\forall k\), for (19). In fact, it was shown in ([15] Theorem 1) that the SDP relaxation under a sumpower constraint always admits a rankone optimal solution when ε _{ k }, ∀k, are sufficiently small. Likewise, we extend the approach to establish the following lemma for (23) under perantenna power constraints:
Lemma 4.
Suppose that for some choice of channel uncertainty bounds \(\boldsymbol {\bar {\epsilon }}=[\bar {\epsilon }_{1},\cdot,\bar {\epsilon }_{K}]^{T}\), the SDP relaxation of (23) is feasible. Define the set
where q _{ m },∀m, are the optimal Lagrange multipliers corresponding to the perantenna power constraints in (23), and \(V(\boldsymbol {\bar {\epsilon }})/(\sum _{m=1}^{M} P_{m})\) is equal to the optimal value for the SDP relaxation of (23) with given \(\boldsymbol {\bar {\epsilon }}\). Then, for any vector \(\boldsymbol {\epsilon }\in \Omega (\boldsymbol {\bar {\epsilon }})\), we have \(\text {rank}(\boldsymbol {X}_{k}^{*})=1, \forall k\).
Proof.
Please see Appendix 3.
On the other hand, for the case of large ε _{ k }, ∀k, the existence of rankone optimal solutions for the SDP relaxation of (23) cannot be provably guaranteed. Hence, the exact optimal solution to (19) may not be constructed from \(\boldsymbol {X}_{k}^{*}\) with possibly a rank greater than one. In this case, randomization is a widely adopted method to obtain a feasible rankone approximate matrix solution from the SDP relaxation. Specifically, a Gaussian randomization strategy [31] can be applied to get a vector \(\boldsymbol {w}_{k}^{*}\) from \(\boldsymbol {X}_{k}^{*}\), ∀k, to nicely approximate the solution to (19). □
4.3 The RobustLPM design
As shown in the previous subsection, when the CSI uncertainty bounds ε _{ k }, ∀k, are small to some extent, the SDP relaxation of RobustSP (23) is globally optimal. However, it remains open whether (19) has an equivalent convex reformulation (i.e., there always exists a tight SDP relaxation) in general. As a compromise, we consider another interesting reformulation based on more conservative robust SINR constraints compared to those in (19).
The robust SINR constraints in (19) can be rewritten as:
for ∥δ _{ k }∥≤ε _{ k }, where \(\boldsymbol {\delta }_{k}=\boldsymbol {h}_{k}  \boldsymbol {\hat {h}}_{k}\), ∀k. Consider the following robust SINR constraints which appear a bit conservative:
for ∥δ _{ k }∥≤ε _{ k }. Since \(\mathcal {R}\left (\boldsymbol {{h}}_{k}^{H}\boldsymbol {w}_{k}\right) \leq \boldsymbol {{h}}_{k}^{H}\boldsymbol {w}_{k}\), the set of w _{ k } defined by (26) is always contained in that by (25).
With the conservative SINR constraints (26), we consider the following robust beamforming problem formulation:
The problem (27) is a semiinfinite SOCP and hence convex, but it is still not easy to be solved efficiently in the current form. Nevertheless, the conservative SINR constraints (26) can be reformulated as LPMs, where the set of each LPM actually forms a convex cone [32]. Furthermore, the recent elegant result in [33] shows that one can construct an LMI to describe a cone of LPM. Resorting to such an LMI representation, the intended beamforming problem (27) can be reformulated as a convex SDP.
To begin with, we define some notations. The ndimensional SOC (which is also termed Lorentz cone) is defined as [33]:
The realvalued 2M×2M matrices \(\boldsymbol {\tilde {D}}_{m}\), ∀m, are defined as
and the realvalued 2K×(2M+1) matrices \(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}})\), ∀k, are Lorentz positive and satisfy that
Given G=[g _{1},·,g _{ P }]^{T} (with P,Q≥3, \(\boldsymbol {g}_{p}\in \mathbb {R}^{Q}\), ∀p=1,..,P), \(\boldsymbol {\hat {A}}(\boldsymbol {G})\) is generated by the P arrow matrices {A(g _{1}),·,A(g _{ P })}, i.e.,
where g _{0}=g _{1}+g _{2}, g _{−1}=g _{1}−g _{2}, g _{ p }=[g _{ p1},g _{ p2},·,g _{ pQ }]^{T}, ∀p, and
With these notations, we can mimic the approach in [32] to show the following proposition.
Proposition 3.
(RobustLPM) The problem (27) can be reformulated as an SDP:
where \(\boldsymbol {\tilde {w}}:=\left [\boldsymbol {w}_{11}^{T},\boldsymbol {w}_{12}^{T},\cdot,\boldsymbol {w}_{K1}^{T},\boldsymbol {w}_{K2}^{T}\right ]^{T}\), \(\boldsymbol {w}_{k1}=\mathcal {R}(\boldsymbol {w}_{k})\), \(\boldsymbol {w}_{k2}=\mathcal {I}(\boldsymbol {w}_{k})\), ∀k, and \({\mathcal {L}}^{\perp }_{2K1,2M}\) denotes a linear subspace as in (58).
Proof.
Please see Appendix 4.
Proposition 3 shows that the robust beamforming problem (27) can be reformulated as an equivalent SDP (29). We highlight that (29) can be efficiently solved using a handy solver (like SeDuMi [29]) with a guaranteed globally optimal solution. This is different in nature from the RobustSP (23), where the original problem (19) is solved through the SDP relaxation technique (again noting that there is no mathematical proof for the zero gap between (19) and the SDP relaxation of (23) in general). Since more conservative SINR constraints are employed, it is clear that \(\hat {\alpha }^{*}(\lambda)\geq \tilde {\alpha }^{*}(\lambda)\), i.e., the proposed RobustLPM might yield a suboptimal solution to (19). However, simulation results in Section 6 show that RobustLPM can always provide an (near)optimal solution.
Before leaving this section, we would like to remark that RobustLPM (29) includes larger size matrices in the LMI constraints than RobustSP (19), which means that the RobustLPM has higher computational complexity than the SDP relaxation of RobustSP (i.e., removing the rankone constraints). However, as a tradeoff, the global optimality for RobustLPM can be achieved while the global optimality for RobustSP cannot be always guaranteed (i.e., there is a positive gap between the SDP relaxation and the problem itself). Both RobustLPM and the relaxed RobustSP can be efficiently solved by interior methods, practically using a software like CVX. According to Boyd and Vandenberghe ([29] Chapter 11.7), the computational complexities of the interiorpoint method for solving RobustLPM and the relaxed RobustSP are \({\mathcal {O}}(K^{7}M^{6.5})\) and \({\mathcal {O}}\left (\sqrt {KM+M^{2}}K^{3}M^{6}\right)\), respectively. □
5 Worstcase weighted sumrate maximization
In this section, we extend the proposed approach to the optimal robust beamforming for the worstcase WSR maximization under perantenna power constraints. Relying on the MP method [5–7, 16, 18, 19], we show that an efficient algorithm can be developed for the worstcase WSR maximization through solving a sequence of robust maxmin SINR problems.
5.1 Problem formulation
With \(\widetilde {\text {SINR}}_{k}(\boldsymbol {W})=\min _{\boldsymbol {h}_{k}\in \mathcal {H}_{k}} \frac {\boldsymbol {h}_{k}^{H}\boldsymbol {w}_{k}}{\sum _{l\neq k}\boldsymbol {h}_{k}^{H}\boldsymbol {w}_{l}^{2}+{\sigma _{k}^{2}}}\) defined in (15), the achievable worstcase rate of user k can be expressed as:
Let μ _{ k } denote the priority weight for user k. An optimization problem with the objective function of maximizing the worstcase WSR under the perantenna power constraints can be formulated as:
The problem (31) is nonconvex due to the heavily coupled mutualinterference terms in its objective function. However, it can be next converted to a standard MP problem over a norm set and can be solved efficiently.
Firstly, we introduce some definitions. A set \({\mathcal {S}}\) is called normal if z ^{′}≤z and \(\boldsymbol {z} \in {\mathcal {S}}\) implies \(\boldsymbol {z}' \in {\mathcal {S}}\); and a set \({\mathcal {S}}\) is called reverse normal if z ^{′}≥z and \(\boldsymbol {z} \in {\mathcal {S}}\) implies \(\boldsymbol {z}' \in {\mathcal {S}}\). A box [a,b] is defined as the set of all z such that a≤z≤b. A vector \(\boldsymbol {y} \in \mathbb {R}_{+}^{K}\) is an upperboundary point of set \({\mathcal {S}}\) if \(\alpha \boldsymbol {y} \in {\mathcal {S}}\), for any scalar α<1, and \(\alpha \boldsymbol {y} \notin {\mathcal {S}}\), ∀α>1. The set of upperboundary points of \({\mathcal {S}}\) is called upperboundary of \({\mathcal {S}}\) and it is denoted by \(\partial ^{+}{\mathcal {S}}\).
Now define the set \( {\mathcal {W}}:=\{\boldsymbol {W}\;\; \sum _{k=1}^{K}\boldsymbol {w}_{k}^{H}\boldsymbol {D}_{m}\boldsymbol {w}_{k} \leq P_{m}, \forall m\}. \) Introducing an auxiliary vector z=[z _{1},…,z _{ K }]^{T}, we can equivalently reformulate (31) as:
where the feasible set is
Since Φ(z) is a strictly increasing function in each entry of z, the optimal solution z ^{∗} for (32) must satisfy: \(z_{k}^{*} = 1+\widetilde {\text {SINR}}_{k}(\boldsymbol {W}^{*})\), ∀k, for a certain \(\boldsymbol {W}^{*} \in {\mathcal {W}}\); and such W ^{∗} is clearly the optimal solution to the original problem (31).
Further define the following two sets
and let \(\boldsymbol {a}(\boldsymbol {W}):=\left [1+\widetilde {\text {SINR}}_{1}(\boldsymbol {W}), \ldots, 1+\widetilde {\text {SINR}}_{K}(\boldsymbol {W})\right ]^{T}\) for any \(\boldsymbol {W} \in {\mathcal {W}}\). Then \({\mathcal {G}}=\cup _{\boldsymbol {W} \in {\mathcal {W}}} [\boldsymbol {0}, \boldsymbol {a}(\boldsymbol {W})]\) is the union of infinitely many normal boxes; clearly, \({\mathcal {G}}\) is normal [34]. Let b:=[b _{1},…,b _{ K }]^{T}, where
It clearly holds: \(1+\widetilde {\text {SINR}}_{k}(\boldsymbol {W}) \leq b_{k}\), ∀k. Therefore, \({\mathcal {G}} \subset [\boldsymbol {0}, \boldsymbol {b}]\) is a compact normal set with nonempty interior. It is also clear that \({\mathcal {H}}\) is a reverse normal set. It then follows that (32) is a standard MP [34]:
where we maximize an increasing function Φ(z) over the intersection of a compact normal set \({\mathcal {G}}\) and a reverse normal set \({\mathcal {H}}\). For the MP, the maximum is attained on \(\partial ^{+}({\mathcal {G}} \cap {\mathcal {H}})\).
5.2 The POA algorithm
Based on the separation property^{3} of normal sets [34], a polyblock outer approximation (POA) algorithm can be employed to efficiently find the globally optimal solution for an MP. For any finite vector set \({\mathcal {T}}:=\{\boldsymbol {v}_{i}, i=1,\ldots, I\}\), the union of all the boxes [0,v _{ i }], ∀i, is a polyblock. The basic idea is to approximate the feasible set \({\mathcal {G}} \cap {\mathcal {H}}\) by a polyblock. Specifically, we construct a nested sequence of polyblocks: \({\mathcal {P}}_{1} \supset {\mathcal {P}}_{2} \supset \cdots \supset {\mathcal {P}}_{n}\supset \cdots \supset {\mathcal {G}} \cap {\mathcal {H}}\) in such a way that \(\max _{\boldsymbol {z} \in {\mathcal {P}}_{n}} \; \Phi (\boldsymbol {z}) ~ \searrow ~ \max _{\boldsymbol {z} \in ({\mathcal {G}} \cap {\mathcal {H}})} \; \Phi (\boldsymbol {z})\), where \({\mathcal {P}}_{n}\) denotes the polyblock generated at the nth iteration and “ ↘” denotes convergence from above.
A vertex \(\boldsymbol {v}_{i} \in {\mathcal {T}}\) is called proper if there does not exist another \(\boldsymbol {v}_{j} \in {\mathcal {T}}\) such that v _{ j }≥v _{ i }. The maximum of an increasing function over a polyblock is attained at one of its proper vertices. Hence, at the nth iteration, the maximizer for Φ(z) over \({\mathcal {P}}_{n}\) is obtained as:
where \({\mathcal {T}}_{n}\) is the (finite) proper vertex set of \({\mathcal {P}}_{n}\). Note that z ^{n} can be simply obtained by searching over the finite entries of \({\mathcal {T}}_{n}\). If \(\boldsymbol {z}^{n} \in ({\mathcal {G}} \cap {\mathcal {H}})\), then it solves (35). Otherwise, we construct the next polyblock \({\mathcal {P}}_{n+1}\) contained in \({\mathcal {P}}_{n} \backslash \{\boldsymbol {z}^{n}\}\) but still enclosing \({\mathcal {G}} \cap {\mathcal {H}}\), and continue the process.
To construct new polyblock \({\mathcal {P}}_{n+1}\) from \({\mathcal {P}}_{n}\), it requires computing the projection point \(\pi _{\mathcal {G}}(\boldsymbol {z}^{n})\), which is defined as the unique point where the halfline from 0 through z meets \(\partial ^{+}{\mathcal {G}}\), i.e., \(\pi _{\mathcal {G}}(\boldsymbol {z}) = \lambda \boldsymbol {z}\) and \(\lambda = \max \{\alpha \;\; \alpha \boldsymbol {z} \in {\mathcal {G}}\}\). Let \(\boldsymbol {y}^{n} =\left [{y^{n}_{1}},\cdot,{y^{n}_{K}}\right ]^{T}:= \pi _{\mathcal {G}}(\boldsymbol {z}^{n})\), and
where e ^{k} is a unit vector with the only nonzero number (i.e., “1”) in the kth entry. Hence, z ^{n,k} can be obtained through replacing \({z}^{n}_{k}\) by \({y^{n}_{k}}\), and y ^{n}≤z ^{n,k}≤z ^{n},∀k.
Let \({\mathcal {T}}_{n+1}\) be the set obtained from \({\mathcal {T}}_{n}\) by replacing the vertex z ^{n} with K new vertices z ^{n,k} and then removing the improper vertices; i.e., \({\mathcal {T}}_{n+1}=({\mathcal {T}}_{n} \backslash \{\boldsymbol {z}^{n}\})\cup \{\boldsymbol {z}^{n,k}\;\;\boldsymbol {z}^{n,k}~\text {is proper}\}\). Since \(\boldsymbol {z}^{*} \in {\mathcal {H}}\), we can further reduce the vertex set \({\mathcal {T}}_{n+1} = {\mathcal {T}}_{n+1} \cap {\mathcal {H}}\). With such a vertex set \({\mathcal {T}}_{n+1}\), we have ([34] Proposition 17):
Lemma 5.
The polyblock \({\mathcal {P}}_{n+1}\) with vertex set \({\mathcal {T}}_{n+1}\) satisfies \(({\mathcal {G}} \cap {\mathcal {H}}) \subset {\mathcal {P}}_{n+1} \subset {\mathcal {P}}_{n}\).
Lemma 5 provides the direction for construction of \({\mathcal {P}}_{n+1}\) from \({\mathcal {P}}_{n}\) to approximate (33) from outside with increasing accuracy. To this end, we need to find \(\boldsymbol {y}^{n} = \pi _{\mathcal {G}}(\boldsymbol {z}^{n})=\lambda ^{n}\boldsymbol {z}^{n}\), which is determined by solving:
This leads to the following maxmin SINR balancing problem:
and its “dual” minmax power problem with given λ ^{n} is
Problem (39) can be solved either by RobustSP in Section 4.2 or by RobustLPM in Section 4.3. The proposed Algorithm 1 can be applied to find the optimal value λ ^{n} for (38).
The proposed POA algorithm to find an ξoptimal solution for (31) is summarized in Algorithm 2. A key requirement for provable convergence is that z is lower bounded by a strictly positive vector [34]. Since z≥1>0 in (32), it readily follows from ([34] Theorem 1) that:
Proposition 4.
Algorithm 2 globally converges to an ξoptimal solution for (31) and (32).
Proposition 4 establishes that Algorithm 2 can yield the optimal beamformers for (31) with guaranteed convergence and global ξoptimality. Note that the worstcase WSR maximization (31) is in fact NPhard. The proposed POA method (i.e., Algorithm 2) is essentially a smart branchandbound approach, which does not have a worstcase polynomialtime complexity [34]. However, extensive numerical examples have shown that this type of algorithm can solve general MP problems of dimensions 10–15 (while the problems of such dimensions are already very hard to solve by the standard approximation tools) [5, 6, 35]. For those small/medium size problems, the complexity with the algorithm may be affordable for practical implementation. On the other hand, for problems with even larger dimensions, the algorithm may be only suitable for benchmarking purposes.
It is worth noting that similar POA algorithms were adopted to find the optimal beamforming designs for the WSR maximization in other different setups [6, 7, 18, 19, 35]. The proposed Algorithm 2 can be also applied to the optimal robust beamforming designs under other general criteria where the objective functions, e.g., the sum of mean square errors or sum of bit error rates, are monotonic functions of the SINRs; see a unified framework [18, 35].
6 Numerical results
6.1 Simulation setting
Consider a MISO downlink system where the BS, equipped with M antennas, serves K singleantenna users. Each transmit antenna has an equal power budget, i.e., P _{ m }=P/M, ∀m, where P is the total power budget at the BS. Assume a normalized channel estimation \(\boldsymbol {\hat {h}}_{k}\sim \mathcal {CN}(\boldsymbol {0},\boldsymbol {I}_{M})\). The CSI uncertainty bound \(\epsilon _{k}=\epsilon \\boldsymbol {\hat {h}}_{k}\\), ∀k, for a given ε in (14). We assume that \(n_{k}\sim \mathcal {CN}(0,\sigma ^{2})\), ∀k, and σ ^{2}=0.1. Define SNR = P/σ ^{2}, and each user has an equal SINR target, i.e., γ _{ k }=SNR/K, ∀k. The tolerance levels are set as δ=0.01 and ξ=0.01 in Algorithms 1 and 2, respectively.
6.2 Properties of α(λ), \(\hat {\alpha }(\lambda)\), and \(\tilde {\alpha }(\lambda)\)
For one given channel realization with M=4 and K = 3, Fig. 2 shows the curves of the perantenna minmax power functions α ^{∗}(λ) in (10) with perfect CSI, \(\tilde {\alpha }^{*}({\lambda })\) in (23) of RobustSP, and \(\hat {\alpha }^{*}(\lambda)\) in (29) of RobustLPM with respect to the SINRtotarget ratio requirement λ. The input SNR is chosen as SNR=10 dB, and the channel uncertainty bound norm is ε=0.2. The monotonicity of α ^{∗}(λ), \(\tilde {\alpha }^{*}({\lambda })\), and \(\hat {\alpha }^{*}(\lambda)\) is clearly shown in Fig. 2. RobustSP and RobustLPM achieve almost the same value., i.e., \(\tilde {\alpha }^{*}({\lambda }) \approx \hat {\alpha }^{*}({\lambda })\). It shows that the values of α ^{∗}(λ) are smaller than those of the robust ones which consider the channel uncertainty. It means that the robust beamforming design requires more power than the beamforming design with perfect CSI under the same SINR constraints to account for the CSI uncertainty. Besides, as discussed in Sections 3 and 4, the optimal beamforming designs can be obtained by solving α ^{∗}(λ)=1, \(\tilde {\alpha }^{*}(\lambda)=1\), or \(\hat {\alpha }^{*}(\lambda)=1\).
6.3 Average minimum SINR
To gauge the average minimum SINR performance of different designs with uncertainty CSI, we realize 100 independent simulation runs. For each simulation run, we first generate \(\boldsymbol {\hat {h}}_{k}\), ∀k, randomly. Based on (13) in Section 4.1, we obtain 1000 true channel realizations \(\boldsymbol {h}_{k}=\boldsymbol {\hat {h}}_{k}+\boldsymbol {\delta }_{k}\) (where \(\\boldsymbol {\delta }_{k}\\leq \epsilon \\boldsymbol {\hat {h}}_{k}\\)), and compute the average minimum SINR over the channel realizations for each input SNR. Then, the SINR results are averaged over the 100 simulation runs. Figures 3 and 4 demonstrate the average minimum SINR performance among K=3 users for M=4 and M=6, respectively, where ε=0.2. It is observed that the proposed Algorithm 1 based on RobustLPM achieves almost the same average minimum SINR performance as that based on RobustSP.
We also evaluate the rankrelaxation method in [16] under perantenna power constraints, where the robust beamformers are obtained using a bisection search with the aid of SDP and rankrelaxation, and all the iterations that return rankone solutions are stored. In Figs. 3 and 4, we observe that RobustSP/RobustLPM significantly outperforms the method of [16] in high SNR regimes. For example, at SNR = 10 dB in Fig. 3, the proposed designs achieve about 4 dB greater SINR gains than the method of [16]. It implies, in high SNR regimes, the tightness of rank relaxation becomes invalid, and thus a randomization is necessary. This is also shown in Lemma 4, when SNR (which is proportional to \(V(\boldsymbol {\bar {\epsilon }})\)) increases, the CSI uncertainty set that guarantees the tightness becomes smaller.
In addition, the performances of NonRobust design, the minimum mean square error (MMSE) design in [4], and the MoorePenrose ZF design in [23] are also included in Figs. 3 and 4. For these three designs, although the true channel is h _{ k }, the calculation of beamformers is processed by regarding the estimated channel \(\boldsymbol {\hat {h}}_{k}\) as the true channel. Note that in order to implement the MoorePenrose ZF beamforming design, we assume that M≥K. The optimal performance under perfect CSI is also provided as an upper bound for all designs. Figures 3 and 4 show that the proposed RobustSP and RobustLPM designs outperform NonRobust in the whole SNR regimes. This is expected because NonRobust regards the estimated channel \(\boldsymbol {\hat {h}}_{k}\) as the true CSI h _{ k } in calculating the beamformers. If the perfect CSI were available, the more “aggressive” NonRobust would have yielded a higher average minimum user SINR than the robust ones (since the robust designs are conservative in the sense that they aim to provide the worsecase guarantees). However, in the presence of the CSI uncertainty, the performance of NonRobust degrades dramatically at most of times since that pretty small perturbations of uncertainty data can make the nominal optimal solution heavily infeasible and thus meaningless.
The proposed robust designs clearly achieve a larger gain in the average minimum SINR performance than the MMSE and MoorePenrose ZF designs as shown in Figs. 3 and 4. For example, at SNR = 0 dB, the RobustSP and the RobustLPM designs with M=4 achieve about a 4 dB gain in the average minimum SINR performance when compared to the MMSE design. Lastly, we observe that the average minimum SINR performance improves when the number of transmit antennas grows. This is because more antennas lead to more accurate beamformers matched with channel characteristics.
Next, we evaluate the average minimum SINR performance of various designs under different CSI uncertainty norm bounds with SNR = 5 dB and K=3. Figure 5 demonstrates the average minimum SINR performance for M=4 and M=6. Again, it is seen that the proposed RobustLPM achieves almost the same performance as the proposed RobustSP. As expected, the average minimum SINR decreases as the channel uncertainty norm bound ε increases. For example, the average minimum SINR of the proposed two designs when ε=0.5 is about 1 dB smaller than those when ε=0.1.
6.4 Average WSR maximization
Figure 6 shows the average WSR values of different designs with ε=0.2, M=4, and K=3. The weights are μ _{1}=0.2, μ _{2}=0.3, and μ _{3}=0.5. The proposed worstcase WSR maximization beamformers are obtained via Algorithm 2 with ξ=0.01. “RobustSP” and “RobustLPM” in Fig. 6 denote the maxmin SINR solutions in Algorithm 2 are obtained through Algorithm 1 based on RobustSP and RobustLPM, respectively. It is observed that the WSR maximization design based on RobustSP slightly outperforms the one based on RobustLPM, where the corresponding WSR gap is smaller than 0.1 bits/s/Hz in the SNR regimes of consideration. For comparison, we also demonstrate the performances of the WSR maximization beamforming designs for which the maxmin SINR solutions are respectively obtained based on the NonRobust design, the MMSE design [4] and the MoorePenrose ZF design [23]. As shown in Fig. 6, the proposed robust WSR maximization designs achieve significantly higher average WSR, compared to the NonRobust, MMSE, and MoorePenrose ZF ones in the medium and high SNR regimes. For example, at an SNR of 10 dB, the proposed RobustSP achieves about 0.2 bits/s/Hz and 0.6 bits/s/Hz more average WSR over the NonRobust design and the MoorePenrose ZF design,respectively.
7 Conclusions
Under the perantenna power constraints, we proposed an efficient approach to find the robust maxmin SINR beamforming designs by solving a sequence of minmax power problems. We developed RobustSP and RobustLPM for the robust minmax power problems. Building on the minmax power solutions, a bisection search algorithm was developed to obtain the robust maxmin SINR beamformers. Using the maxmin SINR solution as a cornerstone, we further proposed an MP method to find the robust beamformers for the worstcase WSR maximization with guaranteed convergence and global optimality. Numerical results demonstrated that the proposed robust designs provide substantial performance improvement over the existing alternatives.
8 Appendices
8.1 Appendix 1: Proof of Lemma 1
Let \(\check {\boldsymbol {W}}=[\check {\boldsymbol {w}}_{1},\cdot,\check {\boldsymbol {w}}_{1}]\) be the optimal solution for (17) with λ>0, such that \(\tilde {\alpha }^{*}(\lambda) = \max _{1 \leq m\leq M} \frac {\left [\sum _{k=1}^{K} \boldsymbol {\check {w}}_{k} \boldsymbol {\check {w}}_{k}^{H} \right ]_{m,m}}{P_{m}}\) and \(\frac {\widetilde {\text {SINR}}_{k}(\boldsymbol {\check {W}})}{\gamma _{k}} \geq \lambda \).
For another λ ^{′}∈[ 0,λ], let β=λ ^{′}/λ. It is clear: 0 < β < 1. we can show that \(\sqrt {\beta }\boldsymbol {\check {W}}\) is in the feasible set of (17) with λ ^{′}. This is because: ∀k,
On the other hand, we have
Therefore, we must have \(\tilde {\alpha }^{*}(\lambda ^{\prime })\leq \beta \tilde {\alpha }^{*}(\lambda)\). It is easy to see that \(\tilde {\alpha }^{*}(\lambda)>0\) for any λ>0. It in turn implies that \(\tilde {\alpha }^{*}(\lambda ^{\prime })\leq \beta \tilde {\alpha }^{*}(\lambda)<\tilde {\alpha }^{*}(\lambda)\) for 0<λ ^{′}<λ.
8.2 Appendix 2: Proof of Lemma 2
Since for all m=1,·,M,
then the beamforming matrix \(\boldsymbol {W}^{*}(\check {\lambda })\) is in the feasible set of (16). This implies:
Let \(\boldsymbol {W}^{*}=[\boldsymbol {w}^{*}_{1},\cdot,\boldsymbol {w}^{*}_{K}]\) be the optimal solution of (16). We can show that W ^{∗} is in the feasible set of (17) with λ ^{∗} as the minimum \(\frac {\widetilde {\text {SINR}}_{k}(\boldsymbol {W})}{\gamma _{k}}\) requirement. This is because \({\lambda }^{*} = \min _{1 \leq k \leq K} \frac {\widetilde {\text {SINR}}_{k}(\boldsymbol {W}^{*})}{\gamma _{k}}\), or equivalently, \(\frac {\widetilde {\text {SINR}}_{k}(\boldsymbol {W}^{*})}{\gamma _{k}} \geq {\lambda }^{*}\), ∀k. On the other hand, we have \(\left [\sum _{k=1}^{K} \boldsymbol {w}^{*}_{k} \left [\boldsymbol {w}^{*}_{k}\right ]^{H} \right ]_{m,m} \leq P_{m}\), ∀m. Therefore, we have
By Lemma 1, \(\tilde {\alpha }^{*}(\lambda)\) is a strictly increasing function of λ>0. The inequality \({\lambda }^{*} \geq \check {\lambda }\) in (41) implies:
We have both (42) and (43) only when all the inequalities are satisfied with equality, i.e., \({\lambda }^{*}=\check {\lambda }\) and \(\boldsymbol {W}^{*}=\boldsymbol {W}^{*}(\check {\lambda })\).
8.3 Appendix 3: Proof of Lemma 4
We first derive the dual of the downlink beamforming problem (23) without rankone constraints. For convenience of analysis, we scale the objective function of (23) with \(\sum _{m=1}^{M} P_{m}\), so that the objective of minimization is the total transmission power \(\alpha \sum _{k=1}^{K} P_{m}\). Therefore, the problem of interest is expressed as:
It is evident that problem (P _{ ε }) and the SDP relaxation (23) have the same feasible region and the same optimal solution. We introduce the following dual variables for the corresponding constraints as shown in Table 1.
Let Q=diag(q _{1},·,q _{ M }) and Φ=diag(P _{1},·,P _{ M }), then the dual of this SDP is given by
Note that the CSI uncertainty bounds ε=[ε _{1},·,ε _{ K }]^{T} and λ are regarded as the given parameters for both the primal problem (P _{ ε }) and its dual (D _{ ε }). Suppose {X _{ k },t _{ k },α} and {Q,G _{ k },d _{ k },β _{ k }} are the optimal solutions of (P _{ ε }) and (D _{ ε }), respectively. Similar to the discussions in [15, 20], we have:

Problem (D _{ ε }) is always strictly feasible;

Suppose the primal problem (P _{ ε }) is feasible, then strong duality holds true for the prime (P _{ ε }) and its dual (D _{ ε });

β _{ k }>0,∀k;

\(\boldsymbol {G}_{k}+\boldsymbol {d}_{k}\boldsymbol {\hat {h}}_{k}^{H}+\boldsymbol {\hat {h}}_{k}\boldsymbol {d}_{k}^{H}+\beta _{k} \boldsymbol {\hat {h}}_{k}\boldsymbol {\hat {h}}_{k}^{H}\succeq \boldsymbol {0},\forall k\);

rank(X _{ k })≥1,∀k.
For any \(\boldsymbol {\epsilon }=[\epsilon _{1},\cdot \epsilon _{k}]^{T}\in \Omega (\boldsymbol {\bar {\epsilon }})\), there is \(\epsilon _{k}\leq \bar {\epsilon }_{k},\forall k\). This follows that any feasible point of (\(P_{\boldsymbol {\bar {\epsilon }}}\)) must be a feasible point of (P _{ ε }). Hence, (P _{ ε }) is solvable. Let V(ε) denote the optimal value of (P _{ ε }) (or its dual (D _{ ε })). It is clear that
Now, to establish rank(X _{ k })=1, we use the complementary slackness condition Tr(X _{ k } Z _{ k })=0 and prove rank(Z _{ k })=M−1. According to the aforementioned properties, we have β _{ k }>0 and
Additionally,
This implies that \(\text {tr}(\frac {1}{\lambda \gamma _{k}}\boldsymbol {G}_{k})<\min _{m}{q_{m}}\), which leads to
Therefore, the rank(Z _{ k }) is calculated as:
where the last inequation holds true due to the fact that \(\text {rank}\left (\left (\frac {1}{\sqrt {\beta _{k}}}\boldsymbol {d}_{k}+\sqrt {\beta }\boldsymbol {\hat {h}}_{k})(\frac {1}{\sqrt {\beta _{k}}}\boldsymbol {d}_{k}+\sqrt {\beta }\boldsymbol {\hat {h}}_{k}\right)^{H}\right)=1\). Recalling the complementary slackness conditions tr(X _{ k } Z _{ k })=0 and rank(X _{ k })≥1,∀k, we obtain rank(Z _{ k })=M−1 and rank(X _{ k })=1,∀k.
8.4 Appendix 4: Proof of Proposition 3
Firstly, we define w _{ k }:=w _{ k1}+j w _{ k2} (i.e., \(\boldsymbol {w}_{k1}=\mathcal {R}(\boldsymbol {w}_{k})\), and \(\boldsymbol {w}_{k2}=\mathcal {I}(\boldsymbol {w}_{k})\)). Likewise, we have \(\boldsymbol {\hat {h}}_{k}:=\boldsymbol {\hat {h}}_{k1}+j\boldsymbol {\hat {h}}_{k2}\) and δ _{ k } :=δ _{ k1}+j δ _{ k2}. Define \(\boldsymbol {W}_{k,1}\!:=[\!\boldsymbol {w}_{11},\cdot,\boldsymbol {w}_{k1,1},\boldsymbol {w}_{k+1,1},\cdot, \boldsymbol {w}_{K1}] \in \mathbb {R}^{M\times (K1)}\) and \(\boldsymbol {W}_{k,2}:=[\boldsymbol {w}_{12},\cdot,\boldsymbol {w}_{k1,2},\boldsymbol {w}_{k+1,2},\cdot, \boldsymbol {w}_{K2}] \in \mathbb {R}^{M\times (K1)}\). Let \(\boldsymbol {\tilde {w}}:=[\boldsymbol {w}_{11}^{T}, \boldsymbol {w}_{12}^{T},\cdot, \boldsymbol {w}_{K1}^{T}, \boldsymbol {w}_{K2}^{T}]^{T}\) and define K real 2K×2M matrices in terms of λ and \(\boldsymbol {\tilde {w}}\) as
and K real 2Kdimension vectors in terms of λ and \(\boldsymbol {\tilde {w}}\) as
The robust SINR constraints (26) can be then written as
Further define K real 2K×(2M+1) matrices as
and \(\boldsymbol {\tilde {\delta }}_{k}:=\left [\begin {array}{lll} 1 &\boldsymbol {\delta }_{k1}^{T} &\boldsymbol {\delta }_{k2}^{T} \end {array}\right ]^{T} \in \mathbb {R}^{2M+1}\), ∀k.
Note that \(\boldsymbol {C}_{k}(\lambda,\boldsymbol {\tilde {w}})\) and \(\boldsymbol {C}_{k}(\lambda,\boldsymbol {\tilde {w}})\) are affine with respect to w _{ k }, ∀k; so are \(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}}), \forall k\). Then (51) becomes
Let
The perantenna power constraints (7) are recast as SOCs:
where we use the fact that \(\boldsymbol {\tilde {D}}_{m}=\boldsymbol {\tilde {D}}_{m}^{\frac {1}{2}}\).
Then (27) is transformed as the following realvalued optimization problem:
In order to solve (55), let us first deal with the robust SOC constraints (i.e., the second group of constraints therein). For a given λ>0, define set _{ k } as
It can be seen that _{ k } contains linear maps taking \(\mathbb {L}^{2M+1}\) to \(\mathbb {L}^{2K}\). In other words, _{ k } is the set \(\mathcal {M}\) of LPMs such that \(\mathcal {M}[\mathbb {L}^{2M+1}]\subset \mathbb {L}^{2K}\).
Lemma 6.
The set _{ k } consists of all Lorentzpositive matrices \(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}})\), i.e.,
Proof.
It immediately follows from the fact that a Lorentz cone is selfdual [29].
Clearly, the set _{ k } is a closed convex cone, and it thus has an LMI description [33]. With such a convex LMI description of _{ k }, we are able to formulate the problem (55) into a standard SDP problem. For that purpose, we need to introduce some notations and facts as follows.
Let \(\mathcal {S}(n)\) and \(\mathcal {A}(n)\) be the spaces of real symmetric and skewsymmetric n×n matrices, respectively. S _{+}(n) is the cone of positive semidefinite matrices in \(\mathcal {S}(n)\). Let \(\mathcal {L}_{P,Q}\) denote the [P Q(P+1)(Q+1)/4]dimension linear space of biquadratic forms [36]:
It is evident that \(\mathcal {L}_{P,Q}\subset \mathcal {S}(PQ)\). The orthogonal complement of \(\mathcal {L}_{P,Q}\) within \(\mathcal {S}(PQ)\) is the [P Q(P−1)(Q−1)/4]dimension subspace \(\mathcal {L}^{\perp }_{P,Q}\), where
It is clear that \(\mathcal {S}(PQ)=\mathcal {L}_{P,Q}\oplus \mathcal {L}^{\perp }_{P,Q}\), where ⊕ denotes the direct sum of vector spaces. By these definitions, it can be seen that:
□
We recall the notation for \(\boldsymbol {\hat {A}}(\cdot)\) in (28) and cite Theorem 5.6 in [33] as a lemma (see also [32], Lemma 3.2):
Lemma 7.
Let min{P,Q}≥3. Then a matrix \(\boldsymbol {G}\in \mathbb {R}^{P\times Q}\) is Lorentzpositive, if and only if there exists \(\boldsymbol {X}\in \mathcal {A}(P1)\otimes \mathcal {A}(Q1)\) such that
Based on (59) and Lemma 6, we can derive an equivalent condition for the Lorentzpositive \(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}})\) in _{ k } as follows.
Proposition 5.
For a given λ>0, \(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}})\) is Lorentz positive if and only if there is \(\boldsymbol {Z}_{k}\in \mathcal {L}^{\perp }_{2K1,2M}\) such that
Proof.
The proof follows immediately from ([32] Proposition 3.3) and the observation in ([32] Equation (27)), and thus we omit it here.
We note that \(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}})\) and \(\boldsymbol {\hat {A}}(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}}))\) (for a given λ>0) are affine with respect to the design variable \(\boldsymbol {\tilde {w}}\), and that \(\mathcal {L}^{\perp }_{2K1,2M}\) is a linear subspace of symmetric matrices with skewsymmetric blocks. It follows that (61) is an implementable LMI description for \(\boldsymbol {B}_{k}(\lambda,\boldsymbol {\tilde {w}})\). Therefore, we obtain the following equivalent convex SDP reformulation of (55):
Now we complete the proof of Proposition 3. □
9 Endnotes
^{1} The proof can be found in our conference version [28].
^{2} It can be ready to extend the results under the assumption of the spherical uncertainty region to the ellipsoidal region case: \(\mathcal {E}_{k}:=\left \{\boldsymbol {\hat {h}}_{k} + {\boldsymbol {\delta }}_{k} \  \ {{\boldsymbol {\delta }}^{H}_{k}\boldsymbol {B}_{k}\boldsymbol {\delta }_{k}} \leq {\epsilon _{k}^{2}} \right \}\), where B _{ k },∀k, are positive definite matrices.
^{3} Analogous to the separation property of convex set, any point z outsides a normal set can be separated from the normal set by a cone congruent to the nonnegative orthant. Thus, a normal set can be approximated as closely as desired by a nested sequence of polyblocks.
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Acknowledgements
This work was supported in part by China Recruitment Program of Global Young Experts, the Program for New Century Excellent Talents in University, National Natural Science Foundation of China under Grant No. 61271223, and the National Science and Technology Major Project of the Ministry of Science and Technology of China under Grant No.2012ZX03001013.
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Wang, F., Huang, Y., Wang, X. et al. Robust beamforming designs for multiuser MISO downlink with perantenna power constraints. J Wireless Com Network 2015, 204 (2015). https://doi.org/10.1186/s1363801504289
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DOI: https://doi.org/10.1186/s1363801504289