Approximation Bounds for Quadratic Optimization with Homogeneous Quadratic Constraints∗

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Approximation Bounds for Quadratic Optimization with Homogeneous Quadratic Constraints

^∗

Zhi-Quan Luo^†, Nicholas D. Sidiropoulos^‡, Paul Tseng^§, and Shuzhong Zhang^¶

June 16, 2006

Abstract

We consider the NP-hard problem of finding a minimum norm vector in n-dimensional real or complex Euclidean space, subject to m concave homogeneous quadratic con- straints. We show that a semidefinite programming (SDP) relaxation for this noncon- vex quadratically constrained quadratic program (QP) provides an O(m²) approxima- tion in the real case, and an O(m) approximation in the complex case. Moreover, we show that these bounds are tight up to a constant factor. When the Hessian of each constraint function is of rank 1 (namely, outer products of some given so-called steer- ing vectors) and the phase spread of the entries of these steering vectors are bounded away from π/2, we establish a certain “constant factor” approximation (depending on the phase spread but independent of m and n) for both the SDP relaxation and a convex QP restriction of the original NP-hard problem. Finally, we consider a re- lated problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints. We show that a SDP relaxation for this nonconvex QP provides an O(1/ ln(m)) approximation, which is analogous to a result of Nemirovski, Roos and Terlaky [14] for the real case.

∗The first author is supported in part by the National Science Foundation, Grant No. DMS-0312416, and by the Natural Sciences and Engineering Research Council of Canada, Grant No. OPG0090391. The second author is supported in part by the U.S. ARO under ERO, Contract No. N62558-03-C-0012, and the EU under U-BROAD STREP, Grant No. 506790. The third author is supported by the National Science Foundation, Grant No. DMS-0511283. The fourth author is supported by Hong Kong RGC Earmarked Grant CUHK418505.

†Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455, U.S.A. (luozq@ece.umn.edu)

‡Department of Electronic and Computer Engineering, Technical University of Crete, 73100 Chania - Crete, Greece. (nikos@telecom.tuc.gr)

§Department of Mathematics, University of Washington, Seattle, Washington 98195, U.S.A.

(tseng@math.washington.edu)

¶Department of Systems Engineering and Engineering Management, The Chinese University of Hong

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1 Introduction

Consider the quadratic optimization problem with concave homogeneous quadratic constraints:

υ_qp := min kzk² s.t. ^X

`∈Ii

|h^H_` z|² ≥ 1, i = 1, ..., m, z ∈ IFⁿ,

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where IF is either IR or IC, k · k denotes the Euclidean norm in IFⁿ, m ≥ 1, each h_` is a given vector in IFⁿ, and I₁, ..., I_m are nonempty, mutually disjoint index sets satisfying I₁ ∪ · · · ∪ I_m = {1, ..., M }. Throughout, the superscript “H” will denote the complex Hermitian transpose, i.e., for z = x + iy, where x, y ∈ IRⁿ and i² = −1, z^H = x^T − iy^T. Geometrically, the above problem (1) corresponds to finding a least norm vector in a region defined by the intersection of the exteriors of m co-centered ellipsoids. If the vectors h₁, ..., h_M are linearly independent, then M equals the sum of the rank of the matrices defining these m ellipsoids. Notice that the problem (1) is easily solved for the case of n = 1, so we assume n ≥ 2.

We assume that ^P_`∈I_ikh_`k 6= 0 for all i, which is clearly a necessary condition for (1) to be feasible. This is also a sufficient condition (since^S^m_i=1{z | ^P_`∈I_i|h^H_` z|² = 0} is a finite union of proper subspaces of IFⁿ, so its complement is nonempty and any point in its complement can be scaled to be feasible for (1)). Thus, the above problem (1) always has an optimal solution (not necessarily unique) since its objective function is coercive, continuous, and its feasible set is nonempty, closed. Notice, however, that the feasible set of (1) is typically nonconvex and disconnected, with an exponential number of connected components exhibiting little symmetry. This is in contrast to the quadratic problems with convex feasible set but nonconvex objective function considered in [13, 14, 22]. Furthermore, unlike the class of quadratic problems studied in [1, 7, 8, 15, 16, 21, 23, 24, 25, 26], the constraint functions in (1) do not depend on z²₁, ..., z_n² only.

Our interest in the nonconvex QP (1) is motivated by the transmit beamforming problem for multicasting applications [20] and by the wireless sensor network localization problem [6]. In the transmit beamforming problem, a transmitter utilizes an array of n transmitting antennas to broadcast information within its service area to m radio receivers, with receiver i ∈ {1, ..., m} equipped with |I_i| receiving antennas. Let h_`, ` ∈ I_i, denote the n × 1 complex steering vector modelling propagation loss and phase shift from the transmitting antennas to the `th receiving antenna of receiver i. Assuming that each receiver performs spatially matched filtering / maximum ratio combining, which is the

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optimal combining strategy under standard mild assumptions, then the constraint X

`∈Ii

|h^H_` z|² ≥ 1

models the requirement that the total received signal power at receiver i must be above a given threshold (normalized to 1). This constraint is also equivalent to a signal-to-noise ratio (SNR) condition commonly used in data communication. Thus, to minimize the total transmit power subject to individual SNR requirements (one at each receiver), we are led to the QP (1). In the special case where each radio receiver is equipped with a single receiving antenna, the problem reduces to [20]:

min kzk²

s.t. |h^H_` z|² ≥ 1, ` = 1, ..., m, z ∈ IFⁿ,

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This problem is a special case of (1) whereby each ellipsoid lies in IFⁿand the corresponding matrix has rank 1.

In this paper, we first show that the nonconvex QP (2) is NP-hard in either the real or the complex case, which further implies the NP-hardness of the general problem (1).

Then, we consider a semidefinite programming (SDP) relaxation of (1) and a convex QP restriction of (2) and study their worst-case performance. In particular, let υ_sdp, υ_cqp and υ_qp denote the optimal values of the SDP relaxation, the convex QP restriction, and the original QP (1), respectively. We establish a performance ratio of υ_qp/υ_sdp = O(m²) for the SDP relaxation in the real case, and we give an example showing that this bound is tight up to a constant factor. Similarly, we establish a performance ratio of υ_qp/υ_sdp= O(m) in the complex case, and we give an example showing the tightness of this bound. We further show that, in the case when the phase spread of the entries of h₁, ..., h_M is bounded away from π/2, the performance ratios υ_qp/υ_sdp and υ_cqp/υ_qp for the SDP relaxation and the convex QP restriction, respectively, are independent of m and n.

In recent years, there have been extensive studies of the performance of SDP relaxations for nonconvex QP. However, to our knowledge, this is the first performance analysis of SDP relaxation for QP with concave quadratic constraints. Our proof techniques also extend to a maximization version of the QP (1) with convex homogeneous quadratic constraints.

In particular, we give a simple proof of a result analogous to one of Nemirovski, Roos and Terlaky [14] (also see [13, Theorem 4.7]) for the real case, namely, the SDP relaxation for this nonconvex QP has a performance ratio of O(1/ ln(m)).

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2 NP-hardness

In this section, we show that the nonconvex QP (1) is NP-hard in general. First, we notice that, by a linear transformation if necessary, the following problem

minimize z^HQz

subject to |z_`| ≥ 1, ` = 1, ..., n, z ∈ IFⁿ,

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is a special case of (1), where Q ∈ IF^n×n is a Hermitian positive definite matrix (i.e., Q Â 0), and z_` denotes the `th component of z. Hence, it suffices to establish the NP- hardness of (3). To this end, we consider a reduction from the NP-complete partition problem: Given positive integers a₁, a₂, ..., a_N, decide whether there exists a subset I of {1, ..., N } satisfying

X

`∈I

a_`= 1 2

XN

`=1

a_`. (4)

Our reductions differ for the real and complex cases. As will be seen, the NP-hardness proof in the complex case¹ is more intricate than in the real case.

2.1 The Real Case

We consider the real case of IF = IR. Let n := N and a := (a₁, . . . , a_N)^T, Q := aa^T + I_n Â 0, where I_ndenotes the n × n identity matrix.

We show that a subset I satisfying (4) exists if and only if the optimization problem (3) has a minimum value of n. Since

z^TQz = |a^Tz|²+ Xn

`=1

|z_`|² ≥ n whenever |z_`| ≥ 1 ∀ `, z ∈ IRⁿ,

we see that (3) has a minimum value of n if and only if there exists a z ∈ IRⁿ satisfying a^Tz = 0, |z_`| = 1 ∀ `.

The above condition is equivalent to the existence of a subset I satisfying (4), with the correspondence I = {` | z_` = 1}. This completes the proof.

1This NP-hardness proof was first presented in an appendix of [20] and is included here for completeness;

also see [26, Proposition 3.5] for a related proof.

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2.2 The Complex Case

We consider the complex case of IF = IC. Let n := 2N + 1 and a := (a₁, . . . , a_N)^T,

A :=

Ã I_N I_N −e_N a^T 0^T_N −¹₂a^Te_N

! , Q := A^TA + I_n Â 0,

where e_N denotes the N -dimensional vector of ones, 0_N denotes the N -dimensional vector of zeros, and I_n and I_N are identity matrices of sizes n × n and N × N , respectively.

We show that a subset I satisfying (4) exists if and only if the optimization problem (3) has a minimum value of n. Since

z^HQz = kAzk²+ Xn

`=1

|z_`|² ≥ n whenever |z_`| ≥ 1 ∀ `, z ∈ ICⁿ,

we see that (3) has a minimum value of n if and only if there exists a z ∈ ICⁿ satisfying Az = 0, |z_`| = 1 ∀ `.

Expanding Az = 0 gives the following set of linear equations:

0 = z_`+ z_{N +`}− z_n, ` = 1, ..., N, (5) 0 =

XN

`=1

a_`z_`−1 2

Ã_N X

`=1

a_`

!

z_n. (6)

For ` = 1, ..., 2N , since |z_`| = |z_n| = 1 so that z_`/z_n = e^iθ^` for some θ_` ∈ [0, 2π), we can rewrite (5) as

cos θ_`+ cos θ_{N +`} = 1,

sin θ_`+ sin θ_{N +`} = 0, ` = 1, ..., N.

These equations imply that θ_` ∈ {−π/3, π/3} for all ` 6= n. In fact, these equations further imply that cos θ_`= cos θ_{N +`}= 1/2 for ` = 1, ..., N , so that

Re Ã_N

X

`=1

a_`z_` z_n −1

2 Ã_N

X

`=1

a_`

!!

= 0.

Therefore, (6) is satisfied if and only if Im

Ã_N X

`=1

a_`z_` z_n −1

2 Ã_N

X

`=1

a_`

!!

= Im Ã_N

X

`=1

a_`z_` z_n

!

= 0,

which is further equivalent to the existence of a subset I satisfying (4), with the corre- spondence I = {` | θ_`= π/3}. This completes the proof.

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3 Performance analysis of SDP relaxation

In this section, we study the performance of an SDP relaxation of (2). Let H_i := ^X

`∈Ii

h_`h^H_` , i = 1, ..., m.

The well-known SDP relaxation of (1) [11, 19] is υ_sdp := min Tr(Z)

s.t. Tr(H_iZ) ≥ 1, i = 1, ..., m, Z º 0, Z ∈ IF^n×n is Hermitian.

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An optimal solution of the SDP relaxation (7) can be computed efficiently using, say, interior-point methods; see [18] and references therein.

Clearly υ_sdp ≤ υ_qp. We are interested in upper bounds for the relaxation performance of the form

υ_qp≤ Cυ_sdp,

where C ≥ 1. Since we assume H_i 6= 0 for all i, it is easily checked that (7) has an optimal solution, which we denote by Z^∗.

3.1 General steering vectors: the real case

We consider the real case of IF = IR. Upon obtaining an optimal solution Z^∗ of (7), we construct a feasible solution of (1) using the following randomization procedure:

1. Generate a random vector ξ ∈ IRⁿ from the real-valued normal distri- bution N (0, Z^∗).

2. Let z^∗(ξ) = ξ/ min

1≤i≤m

q ξ^TH_iξ.

We will use z^∗(ξ) to analyze the performance of the SDP relaxation. Similar procedures have been used for related problems [1, 3, 4, 5, 14]. First, we need to develop two lemmas.

The first lemma estimates the left-tail of the distribution of a convex quadratic form of a Gaussian random vector.

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Lemma 1 Let H ∈ IR^n×n, Z ∈ IR^n×n be two symmetric positive semidefinite matrices (i.e., H º 0, Z º 0). Suppose ξ ∈ IRⁿ is a random vector generated from the real-valued normal distribution N (0, Z). Then, for any γ > 0,

Prob^³ξ^THξ < γE(ξ^THξ)^´≤ max

½√

γ,2(¯r − 1)γ π − 2

¾

, (8)

where ¯r := min{rank (H), rank (Z)}.

Proof. Since the covariance matrix Z º 0 has rank r := rank (Z), we can write Z = U U^T, for some U ∈ IR^n×r satisfying U^TZU = I_r. Let ¯ξ := Q^TU^Tξ ∈ IR^r, where Q ∈ IR^r×r is an orthogonal matrix corresponding to the eigen-decomposition of the matrix

U^THU = QΛQ^T,

for some diagonal matrix Λ = Diag{λ₁, λ₂, ..., λ_r}, with λ₁ ≥ λ₂ ≥ ... ≥ λ_r ≥ 0. Since U^THU has rank at most ¯r, we have λ_i = 0 for all i > ¯r. It is readily checked that ¯ξ has the normal distribution N (0, I_r). Moreover, ξ is statistically identical to U Q¯ξ, so that ξ^THξ is statistically identical to

ξ¯^TQ^TU^THU Q¯ξ = ¯ξ^TΛ¯ξ =

¯

Xr i=1

λ_i|¯ξ_i|². Then, we have

Prob^³ξ^THξ < γE(ξ^THξ)^´ = Prob Ã _¯_r

X

i=1

λ_i|¯ξ_i|² < γE Ã _¯_r

X

i=1

λ_i|¯ξ_i|²

!!

= Prob Ã _¯_r

X

i=1

λ_i|¯ξ_i|² < γ

¯

Xr i=1

λ_i

! .

If λ₁ = 0, then this probability is zero, which proves (8). Thus, we will assume that λ₁> 0. Let ¯λ_i := λ_i/(λ₁+ · · · + λ_r_¯), for i = 1, ..., ¯r. Clearly, we have

¯λ₁+ · · · + ¯λ_¯_r= 1, λ¯₁ ≥ ¯λ₂ ≥ . . . ≥ ¯λ_¯_r≥ 0.

We consider two cases. First, suppose ¯λ₁≥ α, where 0 < α < 1. Then, we can bound the above probability as follows:

Prob^³ξ^THξ < γE(ξ^THξ)^´ = Prob Ã _r_¯

X

i=1

λ¯_i|¯ξ_i|² < γ

!

≤ Prob^³λ¯₁|¯ξ₁|² < γ^´

≤ Prob^³|¯ξ₁|²< γ/α^´ (9)

≤ r2γ

πα,

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where the last step is due to the fact that ¯ξ₁ is a real-valued zero mean Gaussian random variable with unit variance.

In the second case, we have ¯λ₁ < α, so that

λ¯₂+ · · · + ¯λ_r_¯= 1 − ¯λ₁> 1 − α.

This further implies (¯r − 1)¯λ₂ ≥ ¯λ₂+ · · · + ¯λ_r_¯> 1 − α. Hence λ¯₁≥ ¯λ₂ > 1 − α

¯ r − 1.

Using this bound, we obtain the following probability estimate:

Prob^³ξ^THξ < γE(ξ^THξ)^´ = Prob Ã _r_¯

X

i=1

¯λ_i|¯ξ_i|²< γ

!

≤ Prob^³¯λ₁|¯ξ₁|²< γ, ¯λ₂|¯ξ₂|²< γ^´

= Prob^³¯λ₁|¯ξ₁|²< γ^´· Prob^³λ¯₂|¯ξ₂|² < γ^´ (10)

≤

s 2γ π¯λ₁ ·

s 2γ π¯λ₂

≤ 2(¯r − 1)γ π(1 − α).

Combining the estimates for the above two cases and setting α = 2/π, we immediately obtain the desired bound (8).

Lemma 2 Let IF = IR. Let Z^∗º 0 be a feasible solution of (7) and let z^∗(ξ) be generated by the randomization procedure described earlier. Then, with probability 1, z^∗(ξ) is well defined and feasible for (1). Moreover, for every γ > 0 and µ > 0,

Prob µ

1≤i≤mmin ξ^TH_iξ ≥ γ, kξk² ≤ µTr(Z^∗)

¶

≥ 1 − m · max

½√

γ,2(r − 1)γ π − 2

¾

− 1

µ, (11) where r := rank (Z^∗).

Proof. Since Z^∗ º 0 is feasible for (7), it follows that Tr(H_iZ^∗) ≥ 1 for all i = 1, ..., m.

Since E(ξ^TH_iξ) = Tr(H_iZ^∗) ≥ 1 and the density of ξ^TH_iξ is absolutely continuous, the probability of ξ^TH_iξ = 0 is zero, implying that z^∗(ξ) is well defined with probability 1.

The feasibility of z^∗(ξ) is easily verified.

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To prove (11), we first note that E(ξξ^T) = Z^∗. Thus, for any γ > 0 and µ > 0, Prob

µ

1≤i≤mmin ξ^TH_iξ ≥ γ, kξk²≤ µTr(Z^∗)

¶

= Prob^³ξ^TH_iξ ≥ γ ∀ i = 1, ..., m and kξk² ≤ µTr(Z^∗)^´

≥ Prob^³ξ^TH_iξ ≥ γTr(H_iZ^∗) ∀ i = 1, ..., m and kξk² ≤ µTr(Z^∗)^´

= Prob^³ξ^TH_iξ ≥ γE(ξ^TH_iξ) ∀ i = 1, ..., m and kξk² ≤ µE(kξk²)^´

= 1 − Prob^³ξ^TH_iξ < γE(ξ^TH_iξ) for some i or kξk²> µE(kξk²)^´

≥ 1 − Xm i=1

Prob^³ξ^TH_iξ < γE(ξ^TH_iξ)^´− Prob^³kξk²> µE(kξk²)^´

> 1 − m · max

½√

γ,2(r − 1)γ π − 2

¾

− 1 µ,

where the last step uses Lemma 1 as well as Markov’s inequality:

Prob^³kξk² > µE(kξk²)^´≤ 1 µ. This completes the proof.

We now use Lemma 2 to bound the performance of the SDP relaxation.

Theorem 1 Let IF = IR. For the QP (1) and its SDP relaxation (7), we have υ_qp= υ_sdp if m ≤ 2, and otherwise

υ_qp≤ 27m² π υ_sdp.

Proof. By applying a suitable rank reduction procedure if necessary, we can assume that the rank r of the optimal SDP solution Z^∗ satisfies r(r + 1)/2 ≤ m; see e.g. [17]. Thus r <√

2m. If m ≤ 2, then r = 1, implying that Z^∗ = z^∗(z^∗)^T for some z^∗ ∈ IRⁿ and it is readily seen that z^∗ is an optimal solution of (1), so that υ_qp= υ_sdp. Otherwise, we apply the randomization procedure to Z^∗. We also choose

µ = 3, γ = π 4m²

µ 1 − 1

µ

¶₂

= π

9m². Then, it is easily verified using r <√

2m that

√γ ≥ 2(r − 1)γ

π − 2 ∀ m = 1, 2, ...

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Plugging these choices of γ and µ into (11), we see that there is a positive probability (independent of problem size) of at least

1 − m√ γ − 1

µ = 1 −

√π 3 −1

3 = 0.0758...

that ξ generated by the randomization procedure satisfies

1≤i≤mmin ξ^TH_iξ ≥ π

9m² and kξk²≤ 3 Tr(Z^∗).

Let ξ be any vector satisfying these two conditions.² Then, z^∗(ξ) is feasible for (1), so that

υ_qp≤ kz^∗(ξ)k² = kξk²

min_iξ^TH_iξ ≤ 3 Tr(Z^∗)

(π/9m²) = 27m² π υ_sdp, where the last equality uses Tr(Z^∗) = υ_sdp.

In the above proof, other choices of µ can also be used, but the resulting bound seems not as sharp. Theorem 1 suggests that the worst-case performance of the SDP relaxation deteriorates quadratically with the number of quadratic constraints. Below we give an example demonstrating that this bound is in fact tight up to a constant factor.

Example 1: For any m ≥ 2 and n ≥ 2, consider a special instance of (2), corresponding to (1) with |I_i| = 1 (i.e., each H_i has rank 1), whereby

h_`= µ

cos µ`π

m

¶ , sin

µ`π m

¶

, 0, . . . , 0

¶_T

, ` = 1, ...., m.

Let z^∗ = (z₁^∗, . . . , z^∗_n)^T ∈ IRⁿ be an optimal solution of (2) corresponding to the above choice of steering vectors h_`. We can write

(z₁^∗, z^∗₂) = ρ(cos θ, sin θ), for some θ ∈ [0, 2π).

Since {`π/m, ` = 1, ..., m} is uniformly spaced on [0, π), there must exist an integer ` such that

either

¯¯

¯¯θ −`π m −π

2

¯¯

¯¯≤ π 2m or

¯¯

¯¯θ −`π m + π

2

¯¯

¯¯≤ π 2m.

For simplicity, we assume the first case. (The second case can be treated similarly.) Since the last (n − 2) entries of h_` are zero, it is readily checked that

|h^T_`z^∗| = ρ

¯¯

¯¯cos µ

θ −`π m

¶¯¯

¯¯= ρ

¯¯

¯¯sin µ

θ −`π m −π

2

¶¯¯

¯¯≤ ρ

¯¯

¯¯sin µ π

2m

¶¯¯

¯¯≤ ρπ 2m.

2The probability that no such ξ is generated after N independent trials is at most (1−0.0758..)^N, which for N = 100 equals 0.000375.. Thus, such ξ requires relatively few trials to generate.

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Since z^∗ satisfies the constraint |h^T_`z^∗| ≥ 1, it follows that kz^∗k ≥ ρ ≥ 2m|h^T_`z^∗|

π ≥ 2m

π , implying

υ_qp= kz^∗k² ≥ 4m² π² . On the other hand, the positive semidefinite matrix

Z^∗= Diag{1, 1, 0, . . . , 0}

is feasible for the SDP relaxation (7), and it has an objective value of Tr(Z^∗) = 2. Thus, for this instance, we have

υ_qp≥ 2m² π² υ_sdp.

The preceding example and Theorem 1 show that the SDP relaxation (7) can be weak if the number of quadratic constraints is large, especially when the steering vectors h_` are in a certain sense “uniformly distributed” in space.

3.2 General steering vectors: the complex case

We consider the complex case of IF = IC. We will show that the performance ratio of the SDP relaxation (7) improves to O(m) in the complex case (as opposed to O(m²) in the real case). Similar to the real case, upon obtaining an optimal solution Z^∗ of (7), we construct a feasible solution of (1) using the following randomization procedure:

1. Generate a random vector ξ ∈ ICⁿ from the complex-valued normal distribution N_c(0, Z^∗) [2, 26].

2. Let z^∗(ξ) = ξ/ min

1≤i≤m

q ξ^HH_iξ.

Most of the ensuing performance analysis is similar to that of the real case. In particular, we will also need the following two lemmas analogous to Lemmas 1 and 2.

Lemma 3 Let H ∈ IC^n×n, Z ∈ IC^n×n be two Hermitian positive semidefinite matrices (i.e., H º 0, Z º 0). Suppose ξ ∈ ICⁿ is a random vector generated from the complex-valued normal distribution N_c(0, Z). Then, for any γ > 0,

Prob^³ξ^HHξ < γE(ξ^HHξ)^´≤ max

½4

3γ, 16(¯r − 1)²γ²

¾

, (12)

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where ¯r := min{rank (H), rank (Z)}.

Proof. We follow the same notations and proof as for Lemma 1, except for two blanket changes:

matrix transpose → Hermitian transpose, orthogonal matrix → unitary matrix.

Also, ¯ξ has the complex-valued normal distribution N_c(0, I_r). With these changes, we consider the same two cases: ¯λ₁ ≥ α and ¯λ₁ < α, where 0 < α < 1. In the first case, we have similar to (9) that

Prob^³ξ^HHξ < γE(ξ^HHξ)^´≤ Prob^³|¯ξ₁|² < γ/α^´. (13) Recall that the density function of a complex-valued circular normal random variable u ∼ N_c(0, σ²), where σ is the standard deviation, is

1

πσ²e⁻^|u|2^σ2 ∀ u ∈ IC.

In polar coordinates, the density function can be written as f (ρ, θ) = ρ

πσ²e⁻^ρ2^σ2 ∀ ρ ∈ [0, +∞), θ ∈ [0, 2π).

In fact, a complex-valued normal distribution can be viewed as a joint distribution of its modulus and its argument, with the following particular properties: (1) the modulus and argument are independently distributed; (2) the argument is uniformly distributed over [0, 2π); (3) the modulus follows a Weibull distribution with density

f (ρ) =





2ρ

σ²e⁻^ρ2^σ2, if ρ ≥ 0;

0, if ρ < 0, and distribution function

Prob {|u| ≤ t} = 1 − e⁻^σ2^t2. (14) Since ¯ξ₁ ∼ N_c(0, 1), substituting this into (13) yields

Prob^³ξ^HHξ < γE(ξ^HHξ)^´≤ Prob^³|¯ξ₁|²< γ/α^´≤ 1 − e^−γ/α≤ γ/α, where the last inequality uses the convexity of the exponential function.

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In the second case of ¯λ₁< α, we have similar to (10) that

Prob^³ξ^HHξ < γE(ξ^HHξ)^´ ≤ Prob^³¯λ₁|¯ξ₁|²< γ^´· Prob^³λ¯₂|¯ξ₂|² < γ^´

= (1 − e^−γ/¯^λ¹)(1 − e^−γ/¯^λ²)

≤ γ² λ¯₁¯λ₂

≤ (¯r − 1)²γ² (1 − α)² ,

where last step uses the fact that ¯λ₁ ≥ ¯λ₂ ≥ (1 − α)/(¯r − 1). Combining the estimates for the above two cases and setting α = 3/4, we immediately obtain the desired bound (12).

Lemma 4 Let IF = IC. Let Z^∗ º 0 be a feasible solution of (7) and let z^∗(ξ) be generated by the randomization procedure described earlier. Then, with probability 1, z^∗(ξ) is well defined and feasible for (1). Moreover, for every γ > 0 and µ > 0,

Prob µ

1≤i≤mmin ξ^HH_iξ ≥ γ, kξk² ≤ µTr(Z^∗)

¶

≥ 1 − m · max

½4

3γ, 16(r − 1)²γ²

¾

− 1 µ, where r := rank (Z^∗).

Proof. The proof is mostly the same as that for the real case (see Lemma 2). In particular, for any γ > 0 and µ > 0, we still have

Prob µ

¶

≥ 1 − Xm i=1

Prob^³ξ^HH_iξ < γE(ξ^HH_iξ)^´− Prob^³kξk²> µE(kξk²)^´.

Therefore, we can invoke Lemma 3 to obtain Prob

µ

1≤i≤mmin ξ^HH_iξ ≥ γ, kξk²≤ µTr(Z^∗)

¶

≥ 1 − m · max

½4

3γ, 16(r − 1)²γ²

¾

− Prob^³kξk² > µE(kξk²)^´

≥ 1 − m · max

½4

3γ, 16(r − 1)²γ²

¾

− 1 µ, which completes the proof.

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Theorem 2 Let IF = IC. For the QP (1) and its SDP relaxation (7), we have v_sdp = v_qp if m ≤ 3 and otherwise

v_qp≤ 8m · v_sdp.

Proof. By applying a suitable rank reduction procedure if necessary, we can assume that the rank r of the optimal SDP solution Z^∗ satisfies r = 1 if m ≤ 3 and r ≤√

m if m ≥ 4;

see [9, Section 5]. Thus, if m ≤ 3, then Z^∗ = z^∗(z^∗)^H for some z^∗ ∈ ICⁿ and it is readily seen that z^∗ is an optimal solution of (1), so that v_sdp = v_qp. Otherwise, we apply the randomization procedure to Z^∗. By choosing µ = 2 and γ = _4m¹ , it is easily verified using r ≤√

m that

4

3γ ≥ 16(r − 1)²γ² ∀ m = 1, 2, ...

Therefore, it follows from Lemma 4 that Prob

½

¾

≥ 1 − m4 3γ − 1

µ = 1 6.

Then, similar to the proof of Theorem 1, we obtain that with probability of at least 1/6, z^∗(ξ) is a feasible solution of (1) and v_qp≤ kz^∗(ξ)k²≤ 8m · v_sdp.³

The proof of Theorem 2 shows that, by repeating the randomization procedure, the probability of generating a feasible solution with a performance ratio no more than 8m ap- proaches 1 exponentially fast (independent of problem size). Alternatively, a de-randomization technique from theoretical computer science can perhaps convert the above randomization procedure into a polynomial-time deterministic algorithm [12]; also see [14].

Theorem 2 shows that the worst-case performance of SDP relaxation deteriorates lin- early with the number of quadratic constraints. This contrasts with the quadratic rate of deterioration in the real case (see Theorem 1). Thus, the SDP relaxation can yield better performance in the complex case. This is in the same spirit as the recent results in [26]

which showed that the quality of SDP relaxation improves by a constant factor for certain quadratic maximization problems when the space is changed from IRⁿ to ICⁿ. Below we give an example demonstrating that this approximation bound is tight up to a constant factor.

Example 2: For any m ≥ 2 and n ≥ 2, let K = d√

me (so K ≥ 2). Consider a special instance of (2), corresponding to (1) with |I_i| = 1 (i.e., each H_i has rank 1), whereby

h_` = µ

cosjπ

K, sinjπ

Ke^i2kπ^K , 0, . . . , 0

¶_T

with ` = jK − K + k, j, k = 1, ..., K.

3The probability that no such ξ is generated after N independent trials is at most (5/6)^N, which for N = 30 equals 0.00421.. Thus, such ξ requires relatively few trials to generate.

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Hence there are K² complex rank-1 constraints. Let z^∗= (z^∗₁, . . . , z_n^∗)^T ∈ ICⁿbe an optimal solution of (2) corresponding to the above choice of d√

me² steering vectors h_`. By a phase rotation if necessary, we can without loss of generality assume that z₁^∗ is real and write

(z₁^∗, z₂^∗) = ρ(cos θ, sin θe^iψ), for some θ, ψ ∈ [0, 2π).

Since {2kπ/K, k = 1, ..., K} and {jπ/K, j = 1, ..., K} are uniformly spaced in [0, 2π) and [0, π) respectively, there must exist integers j and k such that

¯¯

¯¯ψ −2kπ K

¯¯

¯¯≤ π

K and either

¯¯

¯¯θ − jπ K −π

2

¯¯

¯¯≤ π 2K or

¯¯

¯¯θ −jπ K +π

2

¯¯

¯¯≤ π 2K. Without loss of generality, we assume

¯¯

¯¯θ −jπ K −π

2

¯¯

¯¯≤ π 2K.

Since the last (n − 2) entries of each h_` are zero, it is readily seen that, for ` = jK − K + k,

¯¯

¯Re(h^H_` z^∗)^¯^¯¯ = ρ

¯¯

¯¯cos θ cosjπ

K + sin θ sinjπ K cos

µ

ψ −2kπ K

¶¯¯

¯¯

= ρ

¯¯

¯¯cos µ

θ − jπ K

¶

+ sin θ sinjπ K

µ cos

µ

ψ −2kπ K

¶

− 1

¶¯¯

¯¯

= ρ

¯¯

¯¯sin µ

θ − jπ K −π

2

¶

− 2 sin θ sinjπ K sin²

µKψ − 2kπ 2K

¶¯¯

¯¯

≤ ρ

¯¯

¯¯sin π 2K

¯¯

¯¯+ 2ρ sin² π 2K

≤ ρπ

2K + ρπ² 2K². In addition, we have

¯¯

¯Im(h^H_` z^∗)^¯^¯¯ = ρ

¯¯

¯¯sin θ sinjπ K sin

µ

ψ −2kπ K

¶¯¯

¯¯

≤ ρ

¯¯

¯¯sin µ

ψ −2kπ K

¶¯¯

¯¯

≤ ρ

¯¯

¯¯ψ −2kπ K

¯¯

¯¯≤ ρπ K. Combining the above two bounds, we obtain

¯¯

¯h^H_` z^∗^¯^¯¯≤^¯^¯¯Re(h^H_` z^∗)^¯^¯¯+^¯^¯¯Im(h^H_` z^∗)^¯^¯¯≤ 3ρπ

2K + ρπ² 2K². Since z^∗ satisfies the constraint |h^H_` z^∗| ≥ 1, it follows that

kz^∗k ≥ ρ ≥ 2K²|h^H_` z^∗|

π(3K + π) ≥ 2K² π(3K + π),

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implying

υ_qp= kz^∗k² ≥ 4K⁴

π²(3K + π)² = 4d√ me⁴ π²(3d√

me + π)². On the other hand, the positive semidefinite matrix

Z^∗= Diag{1, 1, 0, . . . , 0}

is feasible for the SDP relaxation (7), and it has an objective value of Tr(Z^∗) = 2. Thus, for this instance, we have

υ_qp≥ 2d√ me⁴ π²(3d√

me + π)² υ_sdp≥ 2m

π²(3 + π/2)² υ_sdp.

The preceding example and Theorem 2 show that the SDP relaxation (7) can be weak if the number of quadratic constraints is large, especially when the steering vectors h_` are in a certain sense “uniformly distributed” in space. In the next subsection, we will tighten the approximation bound in Theorem 2 by considering special cases where the steering vectors are “not too spread out in space”.

3.3 Specially configured steering vectors: the complex case

We consider the complex case of IF = IC. Let Z^∗ be any optimal solution of (7). Since Z^∗ is feasible for (7), Z^∗6= 0. Then

Z^∗= Xr k=1

w_kw_k^H, (15)

for some nonzero w_k ∈ ICⁿ, where r := rank (Z^∗) ≥ 1. By decomposing w_k = u_k+ v_k, with u_k ∈ span{h₁, ..., h_M} and v_k ∈ span{h₁, ..., h_M}^⊥, it is easily checked that ˜Z :=

P_r

k=1u_ku^H_k is feasible for (7) and hI, Z^∗i =

Xr k=1

ku_k+ v_kk² = Xr k=1

(ku_kk²+ kv_kk²) = hI, ˜Zi + Xr k=1

kv_kk².

This implies v_k= 0 for all k, so that

w_k∈ span{h₁, ..., h_M}. (16)

Below we show that the SDP relaxation (7) provides a constant factor approximation to the QP (1) when the phase spread of the entries of h_` is bounded away from π/2.

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Theorem 3 Suppose that

h_` = Xp

i=1

β_i`g_i ∀ ` = 1, ..., M, (17)

for some p ≥ 1, β_i`∈ IC and g_i ∈ ICⁿ such that kg_ik = 1 and g^H_i g_j = 0 for all i 6= j. Then the following results hold.

(a) If Re(β_i`^Hβ_j`) > 0 whenever β_i`^Hβ_j`6= 0, then υ_qp ≤ Cυ_sdp, where

C := max

i,j,` | β_i`^Hβj`6=0

Ã

1 +|Im(β_i`^Hβ_j`)|²

|Re(β_i`^Hβ_j`)|²

!_1/2

. (18)

(b) If β_i`= |β_i`|e^iφ^i`, where

φ_i`∈ [ ¯φ_`− φ, ¯φ_`+ φ] ∀ i, `, for some 0 ≤ φ < π

4 and some ¯φ_` ∈ IR, (19) then Re(β_i`^Hβ_j`) > 0 whenever β_i`^Hβ_j` 6= 0, and C given by (18) satisfies

C ≤ 1

cos(2φ). (20)

Proof. (a) By (16), we have

w_k= Xp

i=1

α_kig_i, for some α_ki∈ IC. This together with (15) yields

hI, Z^∗i = Xr k=1

kw_kk² = Xr k=1

°°

° Xp

i=1

α_kig_i

°°

°

2

= Xr k=1

Xp i=1

|α_ki|² = Xp i=1

λ²_i,

where the third equality uses the orthonormal properties of g₁, ..., g_p, and the last equality uses λ_i :=^¡P^r_k=1|α_ki|²^¢^1/2 = k(α_ki)^r_k=1k.

Let

z^∗ :=

Xp

i=1

λ_ig_i.