decomposable optimisation methods lca reading group, 12/04/2011 dan-cristian tomozei
TRANSCRIPT
![Page 1: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/1.jpg)
Decomposable Optimisation Methods
LCA Reading Group, 12/04/2011Dan-Cristian Tomozei
![Page 2: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/2.jpg)
• Convex function
• Unique minimum over convex domain
Convexity
2
![Page 3: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/3.jpg)
Roadmap• (Sub)Gradient Method• Convex Optimisation crash course• NUM• Basic Decomposition Methods• Implicit Signalling
3
![Page 4: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/4.jpg)
Roadmap• (Sub)Gradient Method• Convex Optimisation crash course• NUM• Basic Decomposition Methods• Implicit Signalling
4
![Page 5: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/5.jpg)
• Unconstrained convex optimisation problem
• If objective is differentiable,
• Else,
• Gain sequence – Constant– Diminishing
(Sub)gradient method
5
![Page 6: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/6.jpg)
Roadmap• (Sub)Gradient Method• Convex Optimisation crash course• NUM• Basic Decomposition Methods• Implicit Signalling
6
![Page 7: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/7.jpg)
• “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
![Page 8: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/8.jpg)
• “Primal” and “dual” formulations
• Karush-Kuhn-Tucker (KKT)
Optimality conditions
Primal variables Dual variables (i.e., Lagrange multipliers)
8
Optimum
![Page 9: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/9.jpg)
Roadmap• (Sub)Gradient Method• Convex Optimisation crash course• NUM• Basic Decomposition Methods• Implicit Signalling
9
![Page 10: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/10.jpg)
• 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
![Page 11: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/11.jpg)
Roadmap• (Sub)Gradient Method• Convex Optimisation crash course• NUM• Basic Decomposition Methods• Implicit Signalling
11
![Page 12: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/12.jpg)
• 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
![Page 13: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/13.jpg)
• 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
![Page 14: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/14.jpg)
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
![Page 15: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/15.jpg)
Roadmap• (Sub)Gradient Method• Convex Optimisation crash course• NUM• Basic Decomposition Methods• Implicit Signalling
15
![Page 16: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/16.jpg)
• 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
![Page 17: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/17.jpg)
Optimisation formulation
• Write Lagrangian
• Primal-dual provably converges to optimum
17
![Page 18: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/18.jpg)
Is it that hard?• Recall
• Dual variables have queue-like evolution!• We already queue packets!
18
![Page 19: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/19.jpg)
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
![Page 20: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/20.jpg)
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
![Page 21: Decomposable Optimisation Methods LCA Reading Group, 12/04/2011 Dan-Cristian Tomozei](https://reader030.vdocuments.mx/reader030/viewer/2022032723/56649d145503460f949e8214/html5/thumbnails/21.jpg)
Questions
21