SDP Relaxation of Quadratic Optimization with Few Homogeneous Quadratic Constraints

Few Homogeneous Quadratic Constraints

Paul Tseng

Mathematics, University of Washington Seattle

LIDS, MIT March 22, 2006

(Joint work with Z.-Q. Luo, N. D. Sidiropoulos, and S. Zhang.) Abstract

This is a talk given at optimization seminar, Oct 2005.

Talk Outline

• Problem description & motivation

• SDP relaxation

• Approximation upper and lower bounds

• Proof idea

• Numerical experience

• A related problem

• Conclusions & open questions

Problem description

υ_qp := min

z∈H kzk² s.t. X

`∈I_i

|h^H_` z|² ≥ 1, i = 1, ..., m,

• h_` 6= 0 ∈ H (H = ICⁿ or IRⁿ), I₁ ∪ · · · ∪ I_m = {1, ..., M }

• z = x + iy (x, y ∈ IRⁿ), z^H = x^T − iy^T

Motivation: Transmit beam forming

SDP Relaxation

•_{. .}Finding a global minimum of QP is NP-hard (reduction from PARTITION).

6

_

• Approximate QP by an “easy” convex optimization problem, a semidefinite program (SDP) relaxation (Lov ´asz ’91, Shor ’87).

SDP Relaxation

Let Z = zz^H (⇐⇒ Z  0, rankZ ≤ 1) H_i = P

`∈I<sub>i</sub> h<sub>`</sub>h^H_` υ_qp = min Tr(Z)

s.t. Tr(H_iZ) = X

`∈I_i

Tr(h_{`</sub>h<sup>H</sup><sub>`} Z) ≥ 1, i = 1, ..., m, Z  0, rankZ ≤ 1.

υ_sdp := min Tr(Z)

s.t. Tr(H_iZ) ≥ 1, i = 1, ..., m, Z  0.

Then

0 ≤ υ_sdp ≤ υ_qp ≤ Cυ^? _sdp (C ≥ 1)

Approximation upper & lower bounds

Theorem 1

2π²m² ≤ C ≤ 27

π m² if H = IRⁿ 1

2(3.6π)²m ≤ C ≤ 8m if H = ICⁿ

Proof sketch

H = IRⁿ

Let Z^∗ be an optimal SDP soln, with rank r ≤ √

2m (such Z^∗ exists).

So Z^∗ = Pr

k=1 z_kz_k^H (z_k ∈ H) Let ζ := Pr

k=1 z_kη_k, η_k ^i.i.d.∼ N (0, 1)

Fact

• E(ζ^HH_iζ) = Tr(H_iZ^∗) ≥ 1 ∀ i

• E(kζk²) = Tr(Z^∗)

• P(ζ^HH_iζ < γ) ≤ √

γ ∀ γ > 0, ∀ i



P(|η_k|² < γ) ≤

q2γ π



• P(kζk² > µTr(Z^∗)) ≤ _µ¹ ∀ µ > 0 (Markov ineq.)

P ζ^HH_iζ ≥ γ, i = 1, ..., m & kζk² ≤ µTr(Z^∗)

≥ 1 −

m

X

i=1

P ζ^HH_iζ < γ
− P kζk² > µTr(Z^∗)

≥ 1 − m√

γ − 1 µ

> 0 if µ = 3, γ = π 9m² so ∃ ζ ∈ <ⁿ such

ζ^HH_iζ ≥ π

9m², i = 1, ..., m kζk² ≤ 3Tr(Z^∗) = 3υ_sdp. Then z :=ˆ √ ^ζ

min_iζ^HH_iζ is a feas. soln of QP, kˆzk² = _min^kζk2

iζ^HH_iζ ≤ _π/(9m^3υsdp₂₎.

Thus υ_qp ≤ kˆzk² ≤ 27

π m²υ_sdp.

Take

n = 2, |I_i| = 1, h_i =  cos(^2π_mi) sin(^2π_mi)



, i = 1, ..., m

• For any QP feas. soln z, ∃ i such |h^H_i z| ≤ π

mkzk ⇒ kzk² ≥ m²

π² ⇒ υ_qp ≥ m²

π²

• Z = I is a feas. soln of SDP, so υ_sdp ≤ Tr(I) = 2 Thus

υ_qp ≥ 1

2π²m² υ_sdp

H = IC

Proof of upper bound is similar to the real case, but with

η_k ^i.i.d.∼ N_c(0, 1) (density e^−|ηk|2 π ) Then P(ζ^HH_iζ < γ) ≤ ⁴₃γ ∀ γ > 0, ∀ i

so

P ζ^HH_iζ ≥ γ, i = 1, ..., m & kζk² ≤ µTr(Z^∗)

≥ 1 −

m

X

i=1

P ζ^HH_iζ < γ
− P kζk² > µTr(Z^∗)

≥ 1 − m4

3γ − 1 µ

> 0 if µ = 2, γ = 1 4m

Proof of lower bound involves a more intricate example.

Improved approximation bound: bounded phase spread

Theorem 2

•

h_` =

p

X

i=1

β<sub>i`</sub>g<sub>i</sub>, ` = 1, ..., M,

for some p ≥ 1, β_i` ∈ IC, g_i ∈ ICⁿ with kg_ik = 1 and g_i^Hg_j = 0 for all i 6= j;

• β_{i`</sub> = |β<sub>i`}|e^iφi` satisfies, for some 0 ≤ φ < ^π₂,

|φ_{i`</sub> − φ<sub>j`}| ≤ φ ∀i, j, ∀`,

then

υ_qp ≤ 1

cos(φ)υ_sdp.

Numerical experience

• For measured VDSL channel data by France Telecom R&D, SDP solution yields nearly doubling of minimum received signal power relative to no precoding.

υ_qp = υ_sdp in over 50% of instances. (SDL ’05)

• Simulation with randomly generated h_` (m = M = 8, n = 4) shows that both the mean and the maximum of the upper bound

kˆxk² υ_sdp

are lower in the H = ICⁿ case (1.14 and 1.8) than the H = IRⁿ case (1.17 and 6.2). Thus, SDP solution is better in the complex case not only in the worst case but also on average.

Maximization QP with convex constraints

υ_qp := max

z∈H kzk² s.t. X

`∈I_i

|h^H_` z|² ≤ 1, i = 1, ..., m,

υ_sdp := max Tr(Z)

s.t. Tr(H_iZ) ≤ 1, i = 1, ..., m, Z  0.

Then

υ_sdp ≥ υ_qp ≥ Cυ^? _sdp (0 < C ≤ 1)

Approximation upper & lower bounds Theorem 3

O

 1 ln(m)



≥ C ≥ 1

4 ln(m) + 2 ln(2) if H = IRⁿ O

 1 ln(m)



≥ C ≥ 1

6 ln(m) + 4 ln(100) if H = ICⁿ

Proof uses P(ζ^HH_iζ > γ) ≤ rank(H_i) e^−γ ∀ γ > 0, ∀ i

Conclusions & Open Questions

1. For norm minimization on IRⁿ (ICⁿ) with m concave quadratic constraints, SDP relaxation yields O(m²) (O(m)) approximation.

2. If phase spread of h₁, ..., h_M are bounded by 0 < φ < ^π₂, then SDP relaxation yields O

1 cos(φ)



approximation.

3. For norm maximization on IRⁿ orCI ⁿ with m convex quadratic constraints, SDP relaxation yields O 

1 ln(m)

 approximation.

Open Questions

1. Can the approximation bounds be improved? Adapt SOS relaxation?

2. For nonconcave/convex constraints, SDP relaxation can be arbitrarily bad (for fixed m, n).

υ_qp := min

(x,y)∈IR²

x² + y²

s.t. y² ≥ 1, x² − M xy ≥ 1, x² + M xy ≥ 1.

(M > 0). Here υ_qp = M + 2 while υ_sdp = 2. Performance of SOS relaxation also worsens with M ↑. Better approximation?

3. A nonhomogeneous QP:

minz∈H z^HH₀z + c^H₀ z

s.t. z^HH_iz + c^H_i z ≥ 1, i = 1, ..., m, can be transformed into a homogeneous QP:

min

(z,t)∈H z^HH₀z + c^H₀ zt

s.t. z^HH_iz + c^H_i zt ≥ 1, i = 1, ..., m, t² = 1.

In the case of m = 2, H₁, H₂  0, c₁ = c₂ = 0, the approximation bound derived from the SDP relaxation of this homogeneous QP is further improved (from 2 to 1.8) by also using the SDP relaxation of

minz∈H z^HH₀z

s.t. z^HH_iz ≥ 1, i = 1, ..., m.

Can this idea be extended?