1. Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei

2. Convex function Unique minimum over convex domain Convexity 2

3. Roadmap (Sub)Gradient Method Convex Optimisation crash course NUM Basic Decomposition Methods Implicit Signalling 3

4. Roadmap (Sub)Gradient Method Convex Optimisation crash course NUM Basic Decomposition Methods Implicit Signalling 4

5. Unconstrained convex optimisation problem If objective is differentiable, Else, Gain sequence – Constant – Diminishing (Sub)gradient method 5

6. Roadmap (Sub)Gradient Method Convex Optimisation crash course NUM Basic Decomposition Methods Implicit Signalling 6

7. “Primal” formulation Convex constraints  unique solution Lagrangian “Dual” function – For all “feasible” points – lower bound – Slater’s condition  zero duality gap Constrained Convex Optimisation 7

8. “Primal” and “dual” formulations Karush-Kuhn-Tucker (KKT) Optimality conditions Primal variablesDual variables (i.e., Lagrange multipliers) 8 Optimum

9. Roadmap (Sub)Gradient Method Convex Optimisation crash course NUM Basic Decomposition Methods Implicit Signalling 9

10. Population of users Concave utility functions (e.g., rates) Typical formulation (e.g., [Kelly97]): – Network flows of rates – Physical links of max capacity – Routing matrix – Dual variables = congestion shadow prices Network Utility Maximisation 10

11. Roadmap (Sub)Gradient Method Convex Optimisation crash course NUM Basic Decomposition Methods Implicit Signalling 11

12. Coupling constraint To decouple – simply write the dual objective Iterative dual algorithm: – Each user computes – Use a gradient method to update dual variables, e.g., Dual Decomposition 12

13. Coupling variable To decouple – consider fixed coupling variable Iterative primal algorithm: – Solve individual problems and get partial optima – Update primal coupling variable using gradient method Primal Decomposition 13

14. Implementation issues Certain problems can be decoupled Dual decomposition  dual algorithm – Primal vars (rates) depend directly on dual vars (prices) – Price adaptation relies on current rates – Always closed form? Primal decomposition – The other way around… Do we really need to keep track of both primal and dual variables? Can duals be “measured” instead? 14

15. Roadmap (Sub)Gradient Method Convex Optimisation crash course NUM Basic Decomposition Methods Implicit Signalling 15

16. Graph Supported rate region Network cost function – Unsupported rate allocation  – Marginal cost positive and strictly increasing Source s wants to send data to receiver r at rate at minimum cost – Supported  min-cut is at least Multipath unicast min-cost live streaming 16

17. Optimisation formulation Write Lagrangian Primal-dual provably converges to optimum 17

18. Is it that hard? Recall Dual variables have queue-like evolution! We already queue packets! 18

19. Implicit Primal-Dual Rate control via Rate on link (i,j) – Increase prop to backlog difference – Decrease prop to marginal cost (measurable – RTT, …) Influence of parameter s – Small  closer optimal allocation, huge queue sizes – Large  manageable queue sizes, optimality trade-off 19

20. Conclusion Finding a fit-all recipe is hard We can handle some cases Specific formulations may lead to nice protocols See also – R. Srikant’s “Mathematics of Internet Congestion Control” – Kelly, Mauloo, Tan - *** – Palomar, Chiang - *** 20

21. Questions 21