Decomposable Optimisation Methods

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

2. Convexity • Convex function • Unique minimum over convex domain

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

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

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

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

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

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

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

10. Network Utility Maximisation • 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

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

12. Dual Decomposition • 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.,

13. Primal Decomposition • 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

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?

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

16. Multipath unicast min-cost live streaming • 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

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

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

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

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 - ***

21. Questions