SDP Relaxation of Quadratic Optimization with Few Homogeneous Quadratic Constraints

Size: px

Start display at page:

Download "SDP Relaxation of Quadratic Optimization with Few Homogeneous Quadratic Constraints"

Lee Jefferson
5 years ago
Views:

1 SDP Relaxation of Quadratic Optimization with 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.

2 SDP RELAXATION OF HOMOGENEOUS QP 1 Talk Outline Problem description & motivation SDP relaxation Approximation upper and lower bounds Proof idea Numerical experience A related problem Conclusions & open questions

3 SDP RELAXATION OF HOMOGENEOUS QP 2 Problem description υ qp := min z H s.t. z 2 l I i h H l z 2 1, i = 1,..., m, h l 0 H (H = IC n or IR n ), I 1 I m = {1,..., M} z = x + iy (x, y IR n ), z H = x T iy T

4 SDP RELAXATION OF HOMOGENEOUS QP 3 Motivation: Transmit beam forming

5 SDP RELAXATION OF HOMOGENEOUS QP 4 SDP Relaxation Finding a global minimum of QP is NP-hard (reduction from PARTITION)... Approximate QP by an easy convex optimization problem, a semidefinite program (SDP) relaxation (Lovász 91, Shor 87).

6 SDP RELAXATION OF HOMOGENEOUS QP 5 SDP Relaxation Let Z = zz H ( Z 0, rankz 1) H i = l I i h l h H l υ qp = min Tr(Z) s.t. Tr(H i Z) = l I i Tr(h l h H l Z) 1, i = 1,..., m, Z 0, rankz 1. υ sdp := min Tr(Z) s.t. Tr(H i Z) 1, i = 1,..., m, Z 0. Then 0 υ sdp υ qp? Cυsdp (C 1)

7 SDP RELAXATION OF HOMOGENEOUS QP 6 Approximation upper & lower bounds Theorem 1 (LSTZ 05): υ qp Cυ sdp where 1 2π 2m2 C 27 π m2 if H = IR n 1 2(3.6π) 2m C 8m if H = ICn

8 SDP RELAXATION OF HOMOGENEOUS QP 7 Proof sketch H = IR n Let Z be an optimal SDP soln, with rank r 2m (such Z exists). So Let Z = r k=1 z kz H k (z k H) ζ := r k=1 z kη k, η k i.i.d. N(0, 1) Fact: E(ζ H H i ζ) = Tr(H i Z ) 1 i E( ζ 2 ) = Tr(Z ) P(ζ H H i ζ < γ) γ γ > 0, i ( P( η k 2 < γ) ) 2γ π P( ζ 2 > µtr(z )) 1 µ µ > 0 (Markov ineq.)

9 SDP RELAXATION OF HOMOGENEOUS QP 8 P ( ζ H H i ζ γ, i = 1,..., m & ζ 2 µtr(z ) ) m 1 P ( ζ H H i ζ < γ ) P ( ζ 2 > µtr(z ) ) i=1 so ζ R n such 1 m γ 1 µ > 0 if µ = 3, γ = π 9m 2 ζ H H i ζ π 9m 2, i = 1,..., m ζ 2 3Tr(Z ) = 3υ sdp. Then ẑ := ζ min i ζ H H i ζ is a feas. soln of QP, ẑ 2 = ζ 2 min i ζ H H i ζ 3υ sdp π/(9m 2 ). Thus υ qp ẑ 2 27 π m2 υ sdp.

10 SDP RELAXATION OF HOMOGENEOUS QP 9 Take n = 2, I i = 1, h i = [ cos( 2π m i) ] sin( 2π m i), i = 1,..., m For any QP feas. soln z, υ qp m2 π 2 i such h H i z π m z z 2 m2 π 2 Z = I is a feas. soln of SDP, so υ sdp Tr(I) = 2 Thus υ qp 1 2π 2m2 υ sdp

11 SDP RELAXATION OF HOMOGENEOUS QP 10 H = IC n Proof of upper bound is similar to the real case, but with η k i.i.d. N c (0, 1) (density e η k 2 ) π Then P(ζ H H i ζ < γ) 4 3 γ γ > 0, i so P ( ζ H H i ζ γ, i = 1,..., m & ζ 2 µtr(z ) ) m 1 P ( ζ H H i ζ < γ ) P ( ζ 2 > µtr(z ) ) i=1 1 m 4 3 γ 1 µ > 0 if µ = 2, γ = 1 4m

12 SDP RELAXATION OF HOMOGENEOUS QP 11 Proof of lower bound involves a more intricate example.

13 SDP RELAXATION OF HOMOGENEOUS QP 12 Improved approximation bound: bounded phase spread Theorem 2 (LSTZ 05): H = IC n. If h l = p β il g i, l = 1,..., M, i=1 for some p 1, β il IC, g i IC n with g i = 1 and g H i g j = 0 for all i j; β il = β il e iφ il satisfies, for some 0 φ < π 2, φ il φ jl φ i, j, l, then υ qp 1 cos(φ) υ sdp.

14 SDP RELAXATION OF HOMOGENEOUS QP 13 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 l (m = M = 8, n = 4) shows that both the mean and the maximum of the upper bound ˆx 2 υ sdp are lower in the H = IC n case (1.14 and 1.8) than the H = IR n 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.

15 SDP RELAXATION OF HOMOGENEOUS QP 14 Maximization QP with convex constraints υ qp := max z H s.t. z 2 l I i h H l z 2 1, i = 1,..., m, υ sdp := max Tr(Z) s.t. Tr(H i Z) 1, i = 1,..., m, Z 0. Then υ sdp υ qp? Cυsdp (0 < C 1)

16 SDP RELAXATION OF HOMOGENEOUS QP 15 Approximation upper & lower bounds Theorem 3 (NRT 99, LSTZ 05): υ qp Cυ sdp where O ( 1 ) ln(m) C 1 4 ln(m) + 2 ln(2) if H = IR n O ( 1 ) ln(m) C 1 6 ln(m) + 4 ln(100) if H = IC n Proof uses P(ζ H H i ζ > γ) rank(h i ) e γ γ > 0, i

17 SDP RELAXATION OF HOMOGENEOUS QP 16 Conclusions & Open Questions 1. For norm minimization on IR n (IC n ) with m concave quadratic constraints, SDP relaxation yields O(m 2 ) (O(m)) approximation. 2. If phase spread of h ( 1,..., h ) M are bounded by 0 < φ < π 2, then SDP relaxation yields O approximation. 1 cos(φ) 3. For norm maximization on ( IR n ) or IC n with m convex quadratic constraints, SDP relaxation yields O approximation. 1 ln(m)

18 SDP RELAXATION OF HOMOGENEOUS QP 17 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 := 2 2 min (x,y) IR 2 x + y s.t. y 2 1, x 2 Mxy 1, x 2 + Mxy 1. (M > 0). Here υ qp = M + 2 while υ sdp = 2. Performance of SOS relaxation also worsens with M. Better approximation?

19 SDP RELAXATION OF HOMOGENEOUS QP A nonhomogeneous QP: min z H zh H 0 z + c H 0 z s.t. z H H i z + c H i z 1, i = 1,..., m, can be transformed into a homogeneous QP: min (z,t) H zh H 0 z + c H 0 zt s.t. z H H i z + c H i zt 1, i = 1,..., m, t 2 = 1. In the case of m = 2, H 1, H 2 0, c 1 = c 2 = 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 min z H zh H 0 z s.t. z H H i z 1, i = 1,..., m. Can this idea be extended?

1. Introduction. Consider the quadratic optimization problem with concave homogeneous quadratic constraints:

1. Introduction. Consider the quadratic optimization problem with concave homogeneous quadratic constraints: APPROXIMATION BOUNDS FOR QUADRATIC OPTIMIZATION WITH HOMOGENEOUS QUADRATIC CONSTRAINTS ZHI-QUAN LUO, NICHOLAS D. SIDIROPOULOS, PAUL TSENG, AND SHUZHONG ZHANG Abstract. We consider the NP-hard problem of