1. Nov 2003Group Meeting #2 Distributed Optimization of Power Allocation in Interference Channel Raul Etkin, Abhay Parekh, and David Tse Spectrum Sharing Group University of California, Berkeley

2. Nov 2003Group Meeting #2 Overview Project goals Interference channel Distributed power allocation problem Cooperative and non-cooperative models Gaussian interference game –Known results –Characterization of Nash equilibria for 2 system case –Extension to more than 2 systems Other approaches 1

3. Nov 2003Group Meeting #2 Project Goals Find strategies to allow multiple systems to share spectrum efficiently Algorithms must be –Distributed –Simple and require minimum cooperation/communication between systems –Robust to non-complying systems –Fair 2

4. Nov 2003Group Meeting #2 Interference Channel M systems, N frequency bands Define System i can use total power P i Background noise AWGN with PSD N 0 1 2 M 1 2 M TxRx............ 3

5. Nov 2003Group Meeting #2 Distributed Power Allocation Problem Power allocation of system i: 4

6. Nov 2003Group Meeting #2 Distributed Power Allocation Problem Power allocation of system i: Rate of system i with Gaussian interference assumption: 4

7. Nov 2003Group Meeting #2 Distributed Power Allocation Problem Power allocation of system i: Rate of system i with Gaussian interference assumption: Want to maximize in a distributed way for some utility function f(.) 4

8. Nov 2003Group Meeting #2 Cooperation Models Cooperative model: –Assume that different systems help each other towards a common goal –Communication ammong systems is costly: want to minimize system interaction Non-cooperative model: –Systems are selfish and try to maximize their own utility function –Systems are rational: will never behave in a way that reduces their own utility –May be used in the cooperative case when system interaction is not possible 5

9. Nov 2003Group Meeting #2 Cooperation Models Cooperative model: –Assume that different systems help each other towards a common goal –Communication ammong systems is costly: want to minimize system interaction Non-cooperative model: –Systems are selfish and try to maximize their own utility function –Systems are rational: will never behave in a way that reduces their own utility –May be used in the cooperative case when system interaction is not possible 5

10. Nov 2003Group Meeting #2 Gaussian Interference Game Non-cooperative Each user tries to maximize R i Optimal strategy: waterfill over noise+interference seen Features/drawbacks: –Distributed, simple, robust, no comm. overhead –Price of anarchy may be infinite (for N 0  0 ) –May have multiple Nash equilibria: cannot predict outcome of game 6

11. Nov 2003Group Meeting #2 Gaussian Interf. Game: Known Results Chung, Cioffi, et. al. analyzed this game under more general assumptions (noise and channel gains can vary over frequency) Have shown: –Nash equilibrium always exist: simple application of a game theory result. –If then Nash equilibrium is unique and stable –If also have unique stable Nash equilibrium 7

12. Nov 2003Group Meeting #2 Gaussian Interf. Game: Known Results Chung, Cioffi, et. al. analyzed this game under more general assumptions (noise and channel gains can vary over frequency) Have shown: –Nash equilibrium always exist: simple application of a game theory result. –If then Nash equilibrium is unique and stable –If also have unique stable Nash equilibrium 7

13. Nov 2003Group Meeting #2 G.I. Game: 2 System Case For M=2 we obtained complete characterization of Nash equilibria If Nash equilibrium is unique and stable. Full spread solution: If have infinite Nash equilibria. Stable solutions have orthogonal power allocations: 8

14. Nov 2003Group Meeting #2 G.I. Game: 2 System Case For M=2 we obtained complete characterization of Nash equilibria If Nash equilibrium is unique and stable. Full spread solution: If have infinite Nash equilibria. Stable solutions have orthogonal power allocations: 8

15. Nov 2003Group Meeting #2 G.I. Game: 2 System Case For M=2 we obtained complete characterization of Nash equilibria If Nash equilibrium is unique and stable. Full spread solution: If have infinite Nash equilibria. Stable solutions have orthogonal power allocations: 8

16. Nov 2003Group Meeting #2 G.I. Game: Multiple Sytems For M > 2 hard to obtain complete characterization of equilibria: –Arbitrary ordering of power allocation update complicates analysis. If equilibrium is unique it must be full spread power allocation. M necessary conditions for stability of full spread solution: Sufficient condition for uniqueness and stability of equilibrium and convergence of iterative waterfilling: 9

17. Nov 2003Group Meeting #2 G.I. Game: Multiple Sytems For M > 2 hard to obtain complete characterization of equilibria: –Arbitrary ordering of power allocation update complicates analysis. If equilibrium is unique it must be full spread power allocation. 9

18. Nov 2003Group Meeting #2 G.I. Game: Multiple Sytems For M > 2 hard to obtain complete characterization of equilibria: –Arbitrary ordering of power allocation update complicates analysis. If equilibrium is unique it must be full spread power allocation. M necessary conditions for stability of full spread solution: 9

19. Nov 2003Group Meeting #2 G.I. Game: Multiple Sytems For M > 2 hard to obtain complete characterization of equilibria: –Arbitrary ordering of power allocation update complicates analysis. If equilibrium is unique it must be full spread power allocation. M necessary conditions for stability of full spread solution: Sufficient condition for uniqueness and stability of equilibrium and convergence of iterative waterfilling: 9

20. Nov 2003Group Meeting #2 Power Minimization Approach Can fix set of required rates and choose power allocations to optimize some utility function. For M=2 and can solve optimization problem in a distributed way by gradient descent. Features/drawbacks: –Choice of f(.) is reasonable for fairness –No communication among systems –Need coordination among systems –Hard to extend for M > 2 –Hard to extend for other utility functions 10

21. Nov 2003Group Meeting #2 Power Minimization Approach Can fix set of required rates and choose power allocations to optimize some utility function. For M=2 and can solve optimization problem in a distributed way by gradient descent. Features/drawbacks: –Choice of f(.) is reasonable for fairness –No communication among systems –Need coordination among systems –Hard to extend for M > 2 –Hard to extend for other utility functions 10

22. Nov 2003Group Meeting #2 Power Minimization Approach Can fix set of required rates and choose power allocations to optimize some utility function. For M=2 and can solve optimization problem in a distributed way by gradient descent. Features/drawbacks: –Choice of f(.) is reasonable for fairness –No communication among systems –Need coordination among systems –Hard to extend for M > 2 –Hard to extend for other utility functions 10

23. Nov 2003Group Meeting #2 Negotiation Based Approach Algorithm: –Fix set of required rates –Start from a feasible point, i.e. orthogonal power allocation is always feasible. –Sequentially let each system vary its power allocation towards minimizing its required power (or some other utility function). –Each system feeds back an accept/reject message for the tentative power allocation. –New power allocation remains in effect if all systems accept it. Features/drawbacks –Fair: all systems have a vote in the optimization process –Feasible for multiple systems –Requires communication and coordination among systems –Usually does not converge to optimal solution 11

24. Nov 2003Group Meeting #2 Negotiation Based Approach Algorithm: –Fix set of required rates –Start from a feasible point, i.e. orthogonal power allocation is always feasible. –Sequentially let each system vary its power allocation towards minimizing its required power (or some other utility function). –Each system feeds back an accept/reject message for the tentative power allocation. –New power allocation remains in effect if all systems accept it. Features/drawbacks –Fair: all systems have a vote in the optimization process –Feasible for multiple systems –Requires communication and coordination among systems –Usually does not converge to optimal solution 11