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Decomposable Optimisation Methods. LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei. Convexity. Convex function Unique minimum over convex domain. Roadmap. (Sub)Gradient Method Convex Optimisation crash course NUM Basic Decomposition Methods Implicit Signalling. Roadmap.
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DecomposableOptimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei
Convexity • Convex function • Unique minimum over convex domain
Roadmap • (Sub)Gradient Method • Convex Optimisationcrash course • NUM • Basic Decomposition Methods • Implicit Signalling
Roadmap • (Sub)Gradient Method • Convex Optimisationcrash course • NUM • Basic Decomposition Methods • Implicit Signalling
(Sub)gradient method • Unconstrained convex optimisation problem • If objective is differentiable, • Else, • Gain sequence • Constant • Diminishing
Roadmap • (Sub)Gradient Method • Convex Optimisationcrash course • NUM • Basic Decomposition Methods • Implicit Signalling
Constrained Convex Optimisation • “Primal” formulation • Convex constraints unique solution • Lagrangian • “Dual” function • For all “feasible” points – lower bound • Slater’s condition zero duality gap
Optimality conditions • “Primal” and “dual” formulations • Karush-Kuhn-Tucker (KKT) Primal variables Dual variables (i.e., Lagrange multipliers) Optimum
Roadmap • (Sub)Gradient Method • Convex Optimisation crash course • NUM • Basic Decomposition Methods • Implicit Signalling
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
Roadmap • (Sub)Gradient Method • Convex Optimisation crash course • NUM • Basic Decomposition Methods • Implicit Signalling
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.,
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
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?
Roadmap • (Sub)Gradient Method • Convex Optimisation crash course • NUM • Basic Decomposition Methods • Implicit Signalling
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
Optimisation formulation • Write Lagrangian • Primal-dual provably converges to optimum
Is it that hard? • Recall • Dual variables have queue-like evolution! • We already queue packets!
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
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 - ***