Non concave network utility maximization - A distributed optimization approach
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The document presents a distributed optimization approach for network utility maximization (NUM) focusing on non-concave user utility functions, particularly in multimedia applications. It introduces a distributed traffic allocation algorithm (DTAA) that enables users to adjust their traffic based on local information and binary feedback, effectively managing congestion and maximizing resource utilization. The work concludes with a convex relaxation of the non-convex NUM problem and suggests future research directions, including numerical simulations and further algorithm development.
Non concave network utility maximization - A distributed optimization approach
- 1. Non-Concave Network Utility Maximization: A Distributed Optimization Approach Presented By: Wasif Hafeez National University of Science and Technology (NUST), Pakistan By: Mahmoud Ashour∗, Jingyao Wang†, Constantino Lagoa∗, Necdet Aybat‡ and Hao Che
- 2. Network Utility Maximization The framework of Network Utility Maximization (NUM) has found many applicatio ns in network rate allocation algorithms and Internet Congestion Control Protocols.
- 3. Concave Utilities Follows from Law of Diminishing Marginal Utilities The problem of Network Utility Maximization in case of concave utilities is essentia lly a convex optimization problem which is solvable efficiently and exactly.
- 4. Non-Concave Utilities In multimedia applications (Video/Audio Streaming) on Internet, the utilities are nonconcave. Nonconvex optimization problem.
- 5. NUM for Non-Concave Utilities We propose a generic framework for the solution of the NUM problem with non- concave user utility functions. We design a sequence of convex relaxations whose solutions converge to that of the original problem. We develop the distributed traffic allocation algorithm (DTAA) that allows users to independently adjust their traffic sending rates and/or redistribute traffic load among multiple routes solely based on available local information and binary feedback from the congested link nodes. The DTAA is shown to be robust to link failures and it is scalable, where the traffic is automatically rerouted in case of a link failure or when new users join the network
- 6. NUM for Non-Concave Utilities The objective is to develop a distributed control protocol which steers the traffic away from congested links so that congestion is avoided and network resource utilization is maximized. In particular, the protocol runs independently in parallel at each source node using local information to allow fully distributed traffic control. The only non-local information needed is whether the forwarding path is congested or not, which is binary feedback from link nodes.
- 7. Notation This work considers network utility functions that can be expressed as a sum of local user utilities, i.e., we maximize subject to network resource constraints and QoS guarantees The traffic is allocated so that no single link in the network is congested. A link l ∈ L is said to be congested if the sum data rate of all sources using that link exceeds its capacity. The network capacity constraints are The optimal traffic allocation that maximizes the network utility is obtained by solving the following optimization problem
- 8. NUM Convex Relaxation We propose a convex relaxation with polynomial like utility functions by leveraging results from the moments approach to polynomial optimization. Instead of solving previously proposed formula, we propose to solve the following semidefinite program: It is worth mentioning that (9) represents a relaxation of (8) when ℓ is even. Nevertheless, a similar result can be obtained when ℓ is odd by slightly modifying the constraints
- 9. Distributed Traffic Allocation Algorithm
- 10. DTAA Results Algorithm 1 summarizes the proposed optimal DTAA The vector xi stores the desired transmission rates of source i over Pi The constraint zi = xi is not satisfied for every iteration. Nevertheless, both zi and xi eventually converge to a consensus as the algorithm keeps running. The information exchange in the network is minimal, where a binary link congestion notification bit is fed back to the source nodes. The algorithm is shown to scale out automatically to accommodate new users joining the network.
- 11. Conclusion This paper addresses the optimization of network utility functions that can be expressed as a sum of local user utilities The utility of each user is a non-concave function of its aggregate data rate. In many practical applications, a non-concave utility function is a better model of the user perceived quality. A convex relaxation of the non-convex NUM problem has been proposed. Furthermore, an optimal decentralized traffic allocation algorithm has been developed. All computations are performed in parallel locally at each user.
- 12. What's Next Future directions include performing numerical simulation of the DTAA on large scale networks to further assess the scalability of the algorithm. Developing a decentralized rate allocation algorithm that allows each node to adapt its rate among any given set of next hops solely based on immediate information from neighboring nodes. Thankyou