Optimization in Wireless Communication

Transcription

1 Zhi-Quan (Tom) Luo Department of Electrical and Computer Engineering University of Minnesota 200 Union Street SE Minneapolis, MN NSF Workshop

2 Challenges Optimization problems from wireless applications often are non-convex distributed in nature dynamic subject to energy constraints... 1

3 Outline Spectrum Management for Multiuser Communication Complexity Asymptotic strong duality Transmit beamforming for multi-group multicast Semidefinite relaxation for nonconvex quadratic optimization Open questions Role of optimization: providing efficient algorithms for distributed maximization with quality assurance; characterizing problem complexity and the structure of optimal solution. 2

4 Spectrum Management for Multiuser Communication With the proliferation of various radio devices and services, multiple systems sharing a common spectrum must coexist Wireline: unbundled DSL Wireless: , Bluetooth, cognitive radios,... Static Spectrum Management: FDMA advantage: orthogonal transmission, zero interference drawback: high system overhead and low bandwidth utilization 3

5 Spectrum Management for Multiuser Communication With the proliferation of various radio devices and services, multiple systems sharing a common spectrum must coexist Wireline: unbundled DSL Wireless: , Bluetooth, cognitive radios,... Static Spectrum Management: FDMA advantage: orthogonal transmission, zero interference drawback: high system overhead and low bandwidth utilization 4

6 Dynamic Spectrum Management Dynamic Spectrum Management: users access a common spectrum simultaneously Each user s performance depends on not only the power allocation (across spectrum) of his own, but also those of other users in the system Proper spectrum management is needed 5

7 Optimal Spectrum Management K users sharing a common resource f Ω; user k s resource allocation s k (f) 0, s k (f)df P k User k s utility: u k = R k (s 1 (f),..., s K (f))df, Ω Example: R k (s 1 (f),..., s K (f)) = log Ω R k ( ) : Lesbegue measurable, non-concave ( 1 + Social optimum: maximizing total weighted sum-rate ) s k (f) σ k (f)+ j k α jk (f)s j (f) max w 1 u w K u K s.t. u 1 = R 1 (s 1 (f),..., s K (f))df Ω. u K = R K (s 1 (f),..., s K (f))df Ω s k (f) 0, s k (f)df P k, k = 1,..., K, Ω (P c ) nonconvex infinite dimensional 6

8 Discretized Spectrum Management Discrete resource set Ω = {1, 2,..., N}; Lebesque measure discrete uniform measure on Ω; user k s resource allocation s n k 0, 1 N s n k N P k User k s utility: u k = 1 N log (1 + n=1 s n k σ n k + j k αn jk sn j Social optimum: maximizing weighted sum-rate ), non-concave max s.t. 1 N 1 N K N ( s n k w k log 1 + σ n n=1 k + j k αn jk sn j N s n k P k, s n k 0, k = 1,..., K, k=1 n=1 ) (P N d ) nonconvex finite dimensional 7

9 Optimal Spectrum Management If α n jk αn kj 1 4 (strong interference on all tones between all users), then optimal spectrum sharing strategy is FDMA (P N d ) is NP-hard even for the two user case; approximately solving (P N d ) is also difficult. The dual of (P N d ), denoted by (DN d strategies ), is easier to solve, especially when restricted to FDMA The ( duality gap between (P N d ) and (DN d O N 1/2). ) is positive, but vanishes asymptotically at rate The continuous primal-dual formulations (P c ) and (D c ) exhibits no duality gap (Lyapunov Theorem). The optimal continuous FDMA spectrum sharing strategies can be approximated arbitrarily well in polynomial time. 8

10 Much More is Needed... Characterizing the structure of optimal spectrum sharing strategy when only some users experience strong interference on some tones. Fast re-optimization when users enter or drop out of the network. Game theoretic models (e.g., jammer) and computation. Iterative water-filling (no good for strong interference case) Distributed optimization with energy efficient message passing. 9

11 Downlink Transmit Beamforming Downlink transmission: basestation has K antennas; m receivers n multicast groups, {G 1,..., G n }, G k G l = k G k = {1,..., m}, G k := G k, n k=1 G k = m. w k : beamforming vector for G k s k : signal sent to group G k transmitted signal: s(t) = n s k (t)w k k=1 10

12 Transmit Beamforming assume each receiver has one antenna, with channel vector h i For ULA, LoS, far field, h i = ( ) 1, e jφ i,..., e j(k 1)φ i where φ i = 2π d λ sin θ i; Vandermonde. signal at receiver i G k : s h i + v i = s k w k h i + l k s l w l h i + v i SINR for receiver i G k w k h i 2 l k w l h i 2 + σi 2 11

13 Problem Description: Transmit Beamforming Transmit beamforming problem: minimize transmit power, subject to QoS constraint for each receiver in each group. minimize subject to n w k 2 k=1 w k h i 2 l k w l h i 2 + σi 2 c i, i G k, k {1,..., n} minimize subject to n w k 2 k=1 w k h i 2 c i w l h i 2 c i σ 2 i, i G k, k {1,..., n} l k Efficiently solvable special cases: ULA; Unicast. NP-hard in general. Use SDP relaxation: strong bounds/empirical results available; also good for joint admission control and beamforming 12

14 Semidefinite Relaxation for Quadratic Optimization Using a fundamental observation: X := xx T X 0, rank(x) = 1, and noting x T P i x = tr (XP i ), the original QCQP: minimize x T P 0 x + r 0 subject to x T P i x + r i 0, i = 1,..., m can be rewritten: minimize tr (XP 0 ) + r 0 subject to tr (XP i ) + r i 0, i = 1,..., m X 0 rank(x) = 1 the only nonconvex constraint is now rank(x) =

15 Convex Relaxation: Semidefinite Relaxation we can directly relax this last constraint rank(x) = 1 to keep only X 0 the resulting program gives a lower bound on the optimal value minimize tr (XP 0 ) + r 0 subject to tr (XP i ) + r i 0, i = 1,..., m X 0 SDP SDP solvable in polynomial time (often in time O(n 3.5 )) From optimal SDP solution X, one can extract a good rank-1 solution ˆx efficiently. For some quadratic optimization problems, one can prove strong bounds on the approximation quality of ˆx. 14

16 Much More is needed... Identify a broader class of nonconvex QPs that are solvable/approximable by SDP relaxation. Distributed and on-line approximation strategies for nonconvex QPs. 15