Complex Quadratic Optimization and Semidefinite Programming

Transcription

1 Complex Quadratic Optimization and Semidefinite Programming Shuzhong Zhang Yongwei Huang August 4; revised April 5 Abstract In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex SDP relaxation problem. In particular, we present an [m 1 cos π m /8π]-apprxomation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of π/4.7854, which is better than the ratio of /π.6366 for its counter-part in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with non-positive off-diagonal elements, then the performance ratio improves to Keywords: Hermitian quadratic functions, approximation ratio, randomized algorithms, complex SDP relaxation. MSC subject classification: 9C, 9C. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong zhang@se.cuhk.edu.hk. Research supported by Hong Kong RGC Earmarked Grants CUHK4174/3E and CUHK44/4E. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong ywhuang@se.cuhk.edu.hk. 1

2 1 Introduction The pioneering work of Goemans and Williamson [8] has caused a great deal of excitement in the field of optimization, as it used a new tool SDP in continuous optimization, through randomization and probabilistic analysis, to yield an excellent approximation ratio for a classical combinatorial optimization problem, known as the max-cut problem. This ground-breaking work has been extended in various ways since its first appearance. Among others, Frieze and Jerrum [6] extended the method to solve the general max-k-cut problem. Bertsimas and Ye [4] introduced another randomization scheme using normal distributions, to achieve the same approximation result as in Goemans and Williamson s original paper [8]. The Bertsimas-Ye analysis makes use of an important result in statistics, which states that the probability of a bivariate -dimensional normally distributed random vector to be in the first orthant can be expressed analytically using elementary functions. This is impossible however, for any dimension higher than three; see [1]. Recently, Goemans and Williamson [9] proposed another novel approach to solve the max-3-cut problem, using the unit circle in the complex plane as a key modelling ingredient. In this paper we show that it is possible to compute the probability of the bivariate complex-valued normally distributed random vector to be in a specific angular region in C see Section. We then consider the following quadratic optimization problem in complex variables: maximizing z H Qz, subject to zk m = 1, where z k is a complex variable and is the k-th component of the vector z, and m is an integer parameter of the model. Thanks to the new probability result developed in Section, we are able to compute the expected quality of a particular randomized solution for solving the above quadratic optimization model. The model of Goemans and Williamson for max-3-cut m = 3 turns out to be a special case of this general model. It is interesting to study the limit of this model; that is, the case where m, and the constraints become z k = 1. It turns out that the problem remains NP-hard. However, the corresponding complex SDP relaxation yields an approximation ratio of π/4.7854, whereas for its counter-part in the real case, the ratio is /π.6366 as shown by Nesterov [11]. If the off-diagonal elements of the objective matrix are real-valued and non-positive, then the approximation ratio is actually This paper is organized as follows. In Section we discuss the computation of the probability for the complex-valued normal distributions. In Section 3 we apply the results developed in Section to solve complex-valued quadratic optimization problems. In particular, Subsection 3.1 discusses the Hermitian quadratic function maximization problem, where the complex decision variables take discrete values. Subsection 3. presents an approximation algorithm for the problem. Subsection 3.3 considers the continuous version of the problem. Subsection 3.4 considers a special case where a sign restriction on the objective matrix is observed. Finally, we conclude the paper in Section 4. Notation. Throughout, we denote by ā the conjugate of a complex number a, by C n the space of n-dimensional complex vectors. For a given vector z C n, z H denotes the conjugate transpose of

3 z. The space of n n real symmetric and complex Hermitian matrices are denoted by S n and H n, respectively. For a matrix Z H n, we write Re Z and Im Z for the real and imaginary part of Z, respectively. Matrix Z being Hermitian implies that Re Z is symmetric and Im Z is skew-symmetric. We denote by S+ n S++ n and H+ n H++ n the cones of real symmetric positive semidefinite positive definite and complex Hermitian positive semidefinite positive definite matrices, respectively. The notation Z means that Z is positive semidefinite positive definite. For two complex matrices Y and Z, their inner product Y Z = Re tr Y H Z = tr [ Re Y T Re Z + Im Y T Im Z ], where tr denotes the trace of a matrix and T denotes the transpose of a matrix. Complex Bivariate Normal Distribution It is well known that the density function of an n-dimensional real-valued multivariate normal distribution is given as follows fx = 1 π n/ det Ω exp 1 x µt Ω 1 x µ, where µ R n is the mean and Ω S n ++ is the covariance matrix. Let us consider a complex-valued normally distributed random variable in C, with the mean value z C and variance σ R +. For more information on the complex-valued normal distributions, we refer the reader to []. Similar as in the real-valued case, its density function can be written as fz = 1 πσ exp z z /σ, z C. We denote the complex-valued normal distribution by N c z, σ with mean z and variance σ. Using Euler s formula, i.e., letting z z = ρe iθ, we have fρ, θ = ρ πσ exp ρ σ, with ρ, θ [, + [, π, where the variable transformation is { Re z z = ρ cos θ Im z z = ρ sin θ. As a matter of terminology, ρ is usually called the modulus of z z, also denoted as z z ; θ is called the argument of z z, denoted as Arg z z. The density of the joint complex-valued normal distribution z = z 1, z,..., z n, with z k C, k = 1,..., n, has the following form fz = 1 π n det Ω exp z µ H Ω 1 z µ, 3

4 where z, µ C n, and Ω H n ++; µ is the mean vector, and Ω is the covariance matrix. Let us denote the above complex-valued normal distribution as N c µ, Ω. The bivariate case is of particular interest to us. Consider a complex-valued, bivariate normal random vector. Suppose that it has zero-mean. Furthermore, suppose that its covariance matrix is [ ] 1 λ Ω = λ 1 where λ C denotes the conjugate of λ C. In particular, let λ = γe iα, and so λ = γe iα. Since Ω, it follows that 1 γ >. Moreover, Ω 1 = 1 1 γ [ 1 γe iα γe iα 1 ]. Then, by letting z 1 = ρ 1 e iθ 1 and z = ρ e iθ, we may rewrite the density function as [ ] H [ ] [ ρ 1 ρ fρ 1, ρ, θ 1, θ = π 1 γ exp 1 ρ 1 e iθ 1 1 γe iα 1 γ ρ e iθ γe iα 1 ρ 1 ρ = π 1 γ exp ρ 1 + ρ ρ 1ρ γ cosα + θ θ 1 1 γ, where the domain of the variables is given as ρ 1, ρ, θ 1, θ [, + [, π. ρ 1 e iθ 1 ρ e iθ ] Now let us further introduce a variable transformation { ρ 1 = ρ cos ξ ρ = ρ sin ξ with the domain ρ, ξ [, + [, π/]. The density function can be further written as fρ, ξ, θ 1, θ = ρ3 cos ξ sin ξ π 1 γ exp ρ γρ cos ξ sin ξ cosα + θ θ 1 1 γ ρ 3 sin ξ = π 1 γ exp ρ ρ γ sin ξ cosα + θ θ 1 1 γ, and the domain is ρ, ξ, θ 1, θ [, + [, π/] [, π. Consider θ1 b < θe 1 π and θb < θe θ 1, θ [θ1 b, θe 1 ] [θb, θe ]. π. Below we shall compute the probability that 4

5 Let us denote P : = Prob {θ1 b θ 1 θ1; e θ b θ θ} e θ e 1 θ e π/ [ ρ 3 sin ξ = π 1 γ exp ρ ρ ] γ sin ξ cosα + θ θ 1 1 γ dρ dξdθ dθ 1 θ b 1 θ b To compute the above integration, we note the following facts: Lemma.1 i For a given a >, it holds that ρ 3 exp aρ dρ = 1 a. ii Suppose that 1 < b < 1 is a given real number. Then, with respect to the variable θ, it holds that sin θ 1 b sin θ dθ = cos θ 1 b 1 b sin θ + b tanθ/ b 1 b arctan + C. 3/ 1 b iii With respect to the variable θ, it holds that [ 1 1 γ cos θ ] γ cos θ arccos γ cos θ + 1 γ cos θ 3/ dθ = 1 1 γ iv With respect to the variable θ, it holds that [ γ sinβ θ arccos γ cosθ β 1 γ cos θ β ] θ + γ sin θ arccos γ cos θ + C. 1 γ cos θ dθ = 1 arccos γ cosθ β + C. Part i of the lemma is straightforward, and the rest of the lemma can be readily verified by differentiation. Applying Lemma.1 i and ii, we get 1 θ e [ 1 θ e π/ P = 4π 1 γ θ1 b θ b = 1 θ γ e [ 1 θ e π/ 4π = 1 γ 4π θ1 b θ b θ e 1 θ e θ b 1 θ b [ sin ξ 1 γ dξ] dθ dθ 1 1 γ sin ξ cosα + θ θ 1 ] sin ξ 1 γ cosα + θ θ 1 sin ξ dξ 1 1 γ cos α + θ θ γ cosα + θ θ 1 arccos γ cosα + θ θ 1 1 γ cos α + θ θ 1 3/ ] dθ dθ 1. dθ dθ 1 5

6 Using Lemma.1 iii we obtain [ P = 1 4π θ1 e θ1θ b e θ b + θ e 1 θ1 b θ e 1 θ b 1 γ sinθ b + α θ 1 arccos γ cosθ b + α θ 1 1 γ cos θ b + α θ 1 γ sinθ e + α θ 1 arccos γ cosθ e + α θ 1 1 γ cos θ e + α θ dθ 1 1 dθ 1, and further using Lemma.1 iv, we have P = θe 1 θb 1 θe θb 4π + 1 [ arccos γ cosθ1 e θ e α arccos γ cosθ1 b θ e α ] + arccos γ cosθ1 b θ b α arccos γ cosθ1 e θ b α. Summarizing, we have proven the following result by a limiting argument. [ Theorem. For the complex-value bivariate normal random vector [ ] [ µ = and Ω = ] 1 γe iα γe iα H+, 1 z 1 z ] N c µ, Ω with it holds that Prob {θ b 1 Arg z 1 θ e 1; θ b Arg z θ e } = θe 1 θb 1 θe θb 4π arccos γ cosθ1 b θ b α [ arccos γ cosθ e 1 θ e α arccos γ cosθ1 e θ b α arccos γ cosθ1 b θ e α ]. 3 Quadratic Programs and Complex SDP Formulations 3.1 Discrete Complex Quadratic Optimization Suppose that Q is a Hermitian matrix. Consider the following quadratic programming problem with discrete decision variables, P max z H Qz s.t. z k {1, ω,..., ω }, k = 1,..., n, where m and ω = e i π m = cos π m + i sin π m. As we shall see later that this model is an extension of the Goemans and Williamson s model for solving the max-3-cut problem; see [9]. 6

7 Denote the optimal value of P to be vp. Consider the following complex-valued mapping F m F m z := m ω 1 ω ω j arccos Re ω j z. For a Hermitian matrix Z with Z kl 1 for all k, l, define the componentwise matrix function F m Z := F m Z kl n n H n. It is easy to verify that F m z = F m z. Therefore, if Z is Hermitian, then so is F m Z. Lemma 3.1 We have 1 = m ω 1 ω Moreover, F m z = z for any z {1, ω,..., ω }. ω j arccos Re ω j. Proof. We observe that Moreover, we have m ω 1 ω = m ω 1 ω = ω 1 ω 8m 4 j ω j = m ω 1 mω ω 1 and ω j arccos cos j m π ω j π 1 j m j ω j 4m jω j = Substituting the above equations into 1 yields the intended result. jω j. 1 m ω 1. 7

8 Suppose z = ω j for some j {1,..., n}. Then, m ω 1 ω = m ω 1 ω = m ω 1 ω = ω j m ω 1 ω = ω j = z. ω j arccos Re ω j z ω j arccos cos j j m π ω j arccos cos j j m π j j= j ω j arccos cos j m π This completes the proof for Lemma 3.1. Hence we can rewrite P as max Q F m zz H s.t. z k {1, ω,..., ω }, k = 1,..., n. Consider the following nonlinear complex semidefinite programming problem SP max Q F m Z s.t. Z kk = 1, k = 1,..., n, Z. Let vsp denote the optimal value of SP. Theorem 3. It holds that vp = vsp. Proof. Let ẑ is optimal to P, then, by Lemma 3.1, Ẑ = ẑẑ H is a feasible solution to SP and F m Ẑ = Ẑ. Therefore, vsp Q FmẐ = Q Ẑ = vp. On the other hand, for every feasible solution Z of SP, we randomly generate a complex vector ξ 8

9 such that ξ N c, Z, and assign z k = σξ k = 1, if Arg ξ k [, 1 m π ω, if Arg ξ k [ 1 m π, m π. ω j, if Arg ξ k [ j j+1 mπ, m π. ω, if Arg ξ k [ m π, π for k = 1,..., n. Suppose that Z kl = γe iα. Then by Theorem., we have Prob {z k = z l ω j, z l = ω r } = Prob {z k = ω j+r, z l = ω r } { = Prob Arg ξ k [ j + r m π, j + r + 1 m π, Arg ξ l [ r m π, r + 1 } m π = 1 m + 1 arccos γ cos j m π α arccos γ cos j 1 π α m arccos γ cos j + 1 π α m for any j, r {, 1,..., m 1}. Therefore, for any given k and l we have = Prob {z k z l = ω j } r= Prob {z k = z l ω j, z l = ω r } = 1 m + m arccos γ cos j m π α arccos γ cos j 1 π α m arccos γ cos j + 1 m π α. 3 9

10 It follows that = E[z k z l ] ω j Prob {z k z l = ω j } = m ω j arccos γ cos j m π α arccos γ cos j 1 π α m arccos γ cos j + 1 π α m = m ω j ω j 1 ω j+1 arccos γ cos j π α m = m ω 1 ω = m ω 1 ω ω j arccos γ cos j π α m By the linearity of mathematical expectation, we get ω j arccos Re ω j Z kl. 4 E[z H Qz] = Q F m Z. Since the solution z so generated is feasible to P, we have vp E[z H Qz] = Q Z, for every feasible solution Z of SP. This combining with vsp vp yields the desired result. In particular, if m = then one can verify that problem P reduces to and problem SP reduces to max x T Qx s.t. x k {±1}, k = 1,..., n, max π Q arcsinx s.t. X kk = 1, k = 1,..., n, X, where arcsinx := [arcsinx kl ] n n. In that case, Theorem 3. specializes to Theorem.9 in Goemans and Williamson [8] or Theorem 1 in Zhang [15]. If m = 3, then P is max z H Qz s.t. z k {1, ω, ω }, k = 1,..., n, 1

11 with ω = e i π 3. In fact, Goemans and Williamson [9] model the max-3-cut problem as M3C max 1 k<l n w klz k z l H z k z l s.t. z k = {1, ω, ω }, k = 1,..., n, and they consider the following complex SDP relaxation max 1 k<l n w kl Re Z kl s.t. Z kk = 1, k = 1,..., n Re Z kl 1/, Re ωz kl 1/, Re ω Z kl 1/, 1 k < l n Z. Let the optimal solution of the SDP relaxation be Z. Then, Theorem 3. asserts that the expected value of the randomized solution based on Z is w kl Re F 3 Zkl 1 k<l n where F 3 z = 9 [ arccos Re z + ωarccos Re ω z + ω arccos Re ωz ]. Since arccosx is a convex function, it follows that Re F 3 Zkl = 9 [ arccos Re Zkl 1 arccos Re ω Zkl + arccos Re ωzkl ] [ 9 arccos Re Zkl arccos 1 Re ωzkl + ω Zkl ] = 9 Further noticing that [arccos Re Z kl arccos 1 Re Z kl ]. [ 9 arccos ] x + 4π min arccos x = x<1 x the approximation ratio of Goemans and Williamson [9] thus follows from the fact that w kl Re F 3 Zkl 1 k<l n { w kl 9 [arccos Re Z kl arccos 1 ]} Re Z kl 1 k<l n.836 w kl Re Zkl 1 k<l n.836 v M3C. The above analysis is due to Goemans and Williamson [9]. Therefore, in this sense Equation 3 is a generalization of Theorem 1 of [9] and our rounding procedure is an extension of the procedure in Section 5.1 of [9]. 11

12 3. Bounds on the Approximation Ratios In this subsection, we investigate approximation algorithms for P with positive semidefinite Q via complex SDP relaxation. Consider the following complex SDP relaxation for P: CSDP max Q Z s.t. Z kk = 1, k = 1,..., n, Z. Suppose that Z is an optimal solution of CSDP. We draw a random vector ξ N c, Z, and generate a feasible solution z C n of P by applying the rounding procedure. In what follows, we wish to establish an approximation ratio α, 1] for the approximation algorithm, i.e., an α such that E[Q zz H ] αq Z, for the randomized solution z. To begin with, we need the following technical lemma, whose proof is given in the appendix of this paper. Lemma 3.3 Suppose that Z H n is positive semidefinite. Then F Z 1 π Z + ZT, and F m Z m 1 cos π m Z for m 3. 8π Therefore, according to Equation 4 and Lemma 3.3, the expectation of the objective value of z can be estimated as E[Q zz H ] = Q F m Z α m Q Z, where { π, if m = α m = m 1 cos π m 8π, if m 3. Hence we arrive at the approximation ratio α m for our randomized algorithm for solving P m. Summarizing, we have the following theorem: Theorem 3.4 Suppose that Q. Then there holds E[Q zz H ] α m vp, where z is obtained by the randomized algorithm and vp is the optimal value of P. In particular, α , α , α , α , and α

13 In the case of m =, CSDP is actually a real SDP problem. According to Lemma 3.3, one asserts that the real version of relaxation problem CSDP yields a π -approximation ratio, which is in accordance with the result of Nesterov [11]. Priori to our result in this subsection Section 3., we learned through private communications that So, Zhang and Ye [1] used a very different technique based on Grothendieck s inequality and obtained the same approximation ratio result for the discrete complex quadratic optimization problem P. We believe that both techniques are interesting and useful in their own rights. 3.3 Continuous Complex Quadratic Optimization By taking the limit, i.e. m, the quadratic optimization model P becomes CP max z H Qz s.t. z k = 1, k = 1,..., n, where Q H n +. In that case, the problem is equivalent to with SCP max Q F Z F z := lim F mz m = 1 4π where γ = z 1 and α = Arg z. π s.t. Z kk = 1, k = 1,..., n, Z e iθ arccos γ cosθ α dθ The applications of Hermitian quadratic optimization models such as P and CP can be found, e.g. in Luo, Luo and Kisialiou [1] for applications in signal processing. Although in [1] the minimization version of the problem was considered, from the viewpoint of optimization both formulations are equivalent see reduction below. Proposition 3.5 Problem CP is strongly NP-hard in general. Proof. The optimization problem in the form of max s.t. z T Az z k C, z k 1, k = 1,..., n 13

14 is called complex programming, and was shown in [13] to be NP-hard in general. We thank André Tits for drawing our attention to complex programming. Problem CP is related to complex programming, but they are not the same: the objective in CP takes the Hermitian form, and is assumed to be positive semidefinite. The proof for Proposition 3.5 to be presented below is due to Tom Luo of Minnesota University, who sketched this proof to us in a private communication. As a first step we shall prove that the following problem min z H Qz s.t. z k = 1, k = 1,..., n, is NP-hard in general, where Q H+. n To this end, we consider a reduction from the following strongly NP-complete matrix partition problem see e.g. [7]; i.e., given a matrix G = [G 1,..., G N ] R M N, decide whether or not a subset of {1,..., N} exists, say I, such that Let the decision vector be Let n = N + 1, and A := G k = 1 k I N G k. z = z, z 1,, z N, z N+1,, z N T C N+1. e N I N I N 1 Ge N G T R M+N n, N where e N R N is the vector of all ones. Let Q := A T A. Next we show that a matrix partition exists is equivalent to the fact that there is z C n with z k = 1 for all k, such that z H Qz =. Clearly, z H Qz = is equivalent to Az = ; that is, = z + z k + z N+k, k = 1,..., N 5 = 1 N N G k z + G k z k. 6 Let z k /z = e iθ k for k = 1,..., N. Using 5 we have cos θ k + cos θ N+k = 1 7 sin θ k + sin θ N+k = 8 where k = 1,..., N. Equations 7 and 8 imply that θ k { π/3, π/3}. This in particular means that cos θ k = cos θ N+k = 1/ for k = 1,..., N. Since Re 1 N N G k + G k z k /z = 1 14 N N G k + G k cos θ k =

15 is always satisfied, 6 is true if and only if Im 1 N G k + N N G k z k /z = G k sin θ k =, which amounts to the existence of a matrix partition. Let λ max be the maximum eigenvalue of Q. By observing that min z H Qz s.t. z k = 1, k = 1,..., n, is equivalent to max z H λ max I Qz s.t. z k = 1, k = 1,..., n, where λ max I Q H+, n the desired result follows. For a given z C with z = γe iα and z = γ 1, we have F z = 1 π 4π e iθ arccos γ cosθ α dθ π = 1 4π eiα e iθ arccos γ cos θ dθ = 1 [ π 4π eiα e iθ arccos γ cos θ dθ = 1 eiα π e iθ π arccosγ cos θ dθ π ] e iθ arccosγ cos θ dθ = 1 π eiα e iθ arcsinγ cos θdθ = 1 π eiα e γ iθ k! cos θ + 4 k k! γ cos θk+1 dθ k + 1 = π 4 γeiα + π k! 4k+1 k! 4 k + 1 γk+1 e iα = π 4 z + π k! 4k+1 k! 4 k + 1 z k z, 9 where the second last step follows from the fact that π sin θcos θ k+1 dθ = and π cos θ k+ dθ = k + 1k 1 1 π, k =, 1,... k + k 15

16 Clearly, if Z H n + then Z T H n +. Furthermore, observe that the Hadamard product of any two positive semidefinite Hermitian matrices remains Hermitian positive semidefinite. Denote A B to k {}}{ be the Hadamard product of A and B, and denote A k to be A A A. It thus follows from 9 that F Z = π 4 Z + π k! 4k+1 k! 4 k + 1 ZT Z k Z π 4 Z. Since Q, we have Q F Z π 4 Q Z. Consider the following complex SDP relaxation for CP CSDP max Q Z s.t. Z kk = 1, k = 1,..., n, Z. Let the optimal value of CP be v CP, the optimal value of CSDP be v CSDP, and Z be an optimal solution. Suppose that an randomized solution z is generated by independently setting z k = e iargξ k for each k = 1,..., n, and ξ N c, Z. Let the expected value of the randomized solution z be vhc. Then vhc π 4 v CSDP π 4 v CP.7854 v CP. Since CP can be viewed as the limit of P as m, it is interesting to observe that the approximation ratio for CP, π 4, is indeed the limit of α m = m 1 cos π m 8π as m. It is also interesting to compare this ratio with that of its real counterpart: RP max x T Qx s.t. x k = 1, k = 1,..., n, where Q is a real positive semidefinite matrix. Nesterov [11] showed that in this case the randomization solution based on the SDP relaxation has the following approximation ratio RSDP max Q X s.t. X kk = 1, k = 1,..., n, X, vhr π v RSDP π v RP.6366 v RP. 16

17 Therefore, the complex SDP relaxation for the complex quadratic optimization problem is more effective than the real SDP relaxation for its real counter-part, in the sense that the former has a slightly better approximation ratio. Remark that similar as the analysis in Nesterov [11], Ye [14], and Zhang [15] for the real case, we can extend all the approximation results to the following more general setting max z H Qz s.t. z 1, z,, z n T F, where F is a closed convex set in R n. The corresponding complex and convex SDP relaxation is max s.t. Q Z diag Z F Z. In particular, if F is a hypercube and Q, then the above π 4 -approximation result also follows from the matrix cube theorem of Ben-Tal, Nemirovski, and Roos [3]. However, our technique appears to be very different in nature. It is also interesting to remark that if we regard CP as an equivalent real quadratic problem max u T, v T Re Q Im Q u Im Q Re Q v s.t. u k + v k = 1, k = 1,..., n, then the approximation ratio obtained that way would be /π, instead of π/4. This shows that the complex SDP relaxation does have an advantage in this particular case. 3.4 Structured Continuous Complex Quadratic Optimization In this subsection, we study a special case of CP with a sign structure on the object matrix, which is parallel to the original real max-cut model studied in [8]: CPS max z H Qz s.t. z k = 1, k = 1,..., n, where we assume that Q = [q jl ] n n S n + and q jl for all 1 j < l n. Using 9 we know that the expected value of the randomized solution based on the complex SDP relaxation is vhc = j<l = j<l q jl Re F Z jl + n q jl π 4 + π j=1 q jj k! n 4k+1 k! 4 k + 1 Z jl k Re Zjl + q jj 1 17 j=1

18 where Z is the optimal solution of the complex SDP relaxation. Define the following real function gy := π 4 + π k! 4k+1 k! 4 k + 1 yk on y [, 1]. We have gy 1 for all y [, 1]. Suppose that x is real, and x y 1. Then, 1 gyx min = min gy + 1 gy = 1 + gyy. x y 1 x x y 1 x 1 + y One computes that Therefore, 1 + gyy min y y.9349 =: β. 1 gyx β βx, for all y [, 1] and x y, or equivalently, gyx 1 β + βx 11 for all y [, 1] and x y. Using 11, we have π 4 + π k! 4k+1 k! 4 k + 1 Z jl k Re Zjl 1 β + βre Z jl. 1 Now we apply 1 in a componentwise fashion to 1, and obtain, thanks to the sign restriction, the following inequalities vhc = j<l j<l q jl π 4 + π q jl 1 β + βre Z jl + n = 1 βe T Qe + βq Z βv CSDP k! n 4k+1 k! 4 k + 1 Z jl k Re Zjl + j=1 q jj βv CP S. 13 j=1 q jj This yields an approximation ratio of.9349 for CPS. 4 Concluding Remarks In this paper we discussed complex quadratic maximization models, denoted as P and CP in the paper, in which the decision variables either take values as unit roots of the equation z m = 1, or are 18

19 assumed to have modulus 1. We established approximation ratios for randomization algorithms for these problems, based on the properties of the complex-valued normal distributions. In particular, the approximation ratio is m 1 cos π m 8π for P, and is π/4 for CP. If the off-diagonal elements of the objective matrix Q are non-positive, then the approximation ratio is improved to Our approach is based on a probability analysis of the complex-valued normally distributed random variables. The same results can also be obtained by different approaches. For example, recently, So, Zhang, and Ye [1] used Grothendieck s inequality and obtained the same m 1 cos π m 8π approximation bound for P, and Ben-Tal, Nemirovski, and Roos [3] established a matrix cube theorem and obtained the π/4 approximation ratio for a model similar to CP. Moreover, Ben-Tal, Nemirovski, and Roos [3] also suggested that the π/4 approximation ratio is a tight bound. However, it remains unknown whether or not α m = m 1 cos π m 8π is a tight bound for P. Related to our models, Charikar and Wirth [5] discussed quadratic maximization models the real case in which Q is not assumed to be positive semidefinite; instead, the diagonals of Q are assumed to be all zeros. They proposed a randomized algorithm for such quadratic maximization model and established an Ω1/ log n-approximation ratio. We plan to extend our analysis to such models in the future. Acknowledgement: We would like to thank Tom Luo, Anthony So, Yinyu Ye, and Jiawei Zhang for stimulating discussions on the subject. A Proof of Lemma 3.3 Consider F m z = m ω ω 1 ω j arccos Re ω j z = m1 cos π m 4π e iα e iθ j arccos γ cos θ j, where z = γe iα, ω = e i π m and θ j = j mπ α for j =,..., m 1. 19

20 Since arccos x = π arcsin x, we have F m z = m1 cos π m 4π e iα e iθ j π 4 + π arcsinγ cos θ j + arcsinγ cos θ j = m1 cos π m 4π e iα πe iθ j arcsinγ cos θ j + e iθ j arcsinγ cos θ j = m1 cos π m 4π e iα πe iθ j γ cos θ j + πe iθ j arcsinγ cos θ j γ cos θ j +e iθ j arcsinγ cos θ j. Set I 1 = γe iα I = e iα I 3 = e iα e iθ j cos θ j e iθ j arcsinγ cos θ j γ cos θ j e iθ j arcsinγ cos θ j. Thus, we shall have F m z = m1 cos π m 4π I 1 + I + I 3 /π. 14 Let us now treat these items one by one. First, we note that I 1 = γe iα = γeiα = γeiα m + = e iθ j cos θ j e iθ j e iθ j + e iθ j e i 4π m j e iα γe iα m = mz/, if m 3 γe iα + γe iα = z + z, if m =. Let us denote a n = n!, n =, 1,... Then we have the Taylor expansion arcsint = n n! n+1

21 n= a nt n+1, and so Let us denote I = e iα = = = n=1 n=1 n=1 e iθ j a n γ n+1 cos θ j n+1 n=1 a n n+1 γn+1 e iα a n n+1 γn+1 e iα a n n+1 γn+1 e iα n+1 k= e iθ j n+1 n+1 e iθ j + e iθ j n + 1 e iθ jn+ k k k= n + 1 e i π m n+ kj e iαn+ k. k b k = e i π m kj where k is an integer number. Obviously, b k is either or m. In particular, if m is even and k is odd, then b k =. We further obtain that Similarly, we have I = I 3 = e iα = e iα = = e iα n=1 n=1 a n n+1 γn+1 e iα a n n+1 n+1 k= e iθ j arcsinγ cos θ j s=,t= s=,t= = e iα = s=,t= s=,t= a s a t n+1 k= n + 1 k n + 1 a s a t γ s+t+ cos θ j s+t+ s+t+ γs+t+ a s a t a s a t s+t+ s+t+ γs+t+ s+t+ k= s+t+ k= s+t+ k= s + t + k b n+ k e iαn+ k b n+ k z n+1 k z k. s + t + k s + t + If m is even, then I 3 = since b s+t+3 k = in each term. k k e iθ js+t+3 k b s+t+3 k e iαs+t+3 k b s+t+3 k z s+t+ k z k. 1

22 Observing that the Hadamard product of any two positive semidefinite Hermitian matrices remains Hermitian positive semidefinite, it follows from 14 that F m Z = m 1 cos π m Z + m 1 cos π m 8π 4π [ n=1 Z k Z T n+1 k] + m 1 cos π m 4π b s+t+3 k Z k Z T s+t+ k], n+1 a n n+1 k= s=,t= [ a s a t s+t+ n + 1 k s+t+ k= b n+ k s + t + k for m 3 while a similar expansion of F Z can be obtained easily. Since Z and Z T, hence we get F m Z m 1 cos π m Z, 8π for m 3 and F Z 1 π Z + ZT. This completes the proof. References [1] I.G. Abrahamson. Orthant probability for the quadrivariate normal distribution. The Annals of Mathematical Statistics 35: , [] H.H. Andersen. Linear and graphical models for the multivariate complex normal distribution. Springer-Verlag, [3] A. Ben-Tal, A. Nemirovski and C. Roos. Extended matrix cube theorems with applications to µ-theory in control. Mathematics of Operations Research 8: , 3. [4] D. Bertsimas and Y. Ye. Semidefinite relaxations, multivarite normal distribution, and order statistics. In D.Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 3, pages Kluwer Academic Publishers, [5] M. Charika and A. Wirth. Maximizing quadratic programs: extending Grothendieck s inequality. Proc. 45th FOCS, 4. [6] A. Frieze and M. Jerrum. Improved approximation algorithms for MAX-k-CUT and Max BI- SECTION. Algorithmica 18: 67 81, [7] M.R. Garey and D.S. Johnson. Computers and Intractability. A guide to the theory of NPcompleteness. W.H. Freeman and Company, 1979.

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