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Register a new userA Distributed Newton Method for Network Utility Maximization
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Added by
Ermin Wei
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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Most existing work uses dual decomposition and subgradient methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This work develops an alternative distributed Newton-type fast converging algorithm for solving network utility maximization problems with self-concordant utility functions. By using novel matrix splitting techniques, both primal and dual updates for the Newton step can be computed using iterative schemes in a decentralized manner with limited information exchange. Similarly, the stepsize can be obtained via an iterative consensus-based averaging scheme. We show that even when the Newton direction and the stepsize in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing subgradient methods based on dual decomposition.
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The problem of minimizing a sum of local convex objective functions over a networked system captures many important applications and has received much attention in the distributed optimization field. Most of existing work focuses on development of fast distributed algorithms under the presence of a central clock. The only known algorithms with convergence guarantees for this problem in asynchronous setup could achieve either sublinear rate under totally asynchronous setting or linear rate under partially asynchronous setting (with bounded delay). In this work, we built upon existing literature to develop and analyze an asynchronous Newton based approach for solving a penalized version of the problem. We show that this algorithm converges almost surely with global linear rate and local superlinear rate in expectation. Numerical studies confirm superior performance against other existing asynchronous methods.
We consider the Network Utility Maximization (NUM) problem for wireless networks in the presence of arbitrary types of flows, including unicast, broadcast, multicast, and anycast traffic. Building upon the recent framework of a universal control policy (UMW), we design a utility optimal cross-layer admission control, routing and scheduling policy, called UMW+. The UMW+ policy takes packet level actions based on a precedence-relaxed virtual network. Using Lyapunov optimization techniques, we show that UMW+ maximizes network utility, while simultaneously keeping the physical queues in the network stable. Extensive simulation results validate the performance of UMW+; demonstrating both optimal utility performance and bounded average queue occupancy. Moreover, we establish a precise one-to-one correspondence between the dynamics of the virtual queues under the UMW+ policy, and the dynamics of the dual variables of an associated offline NUM program, under a subgradient algorithm. This correspondence sheds further insight into our understanding of UMW+.
Distributed and iterative network utility maximization algorithms, such as the primal-dual algorithms or the network-user decomposition algorithms, often involve trajectories where the iterates may be infeasible, convergence to the optimal points of relaxed problems different from the original, or convergence to local maxima. In this paper, we highlight the three issues with iterative algorithms. We then propose a distributed and iterative algorithm that does not suffer from the three issues. In particular, we assert the feasibility of the algorithms iterates at all times, convergence to the global maximum of the given problem (rather than to global maximum of a relaxed problem), and avoidance of any associated spurious rest points of the dynamics. A benchmark algorithm due to Kelly, Maulloo and Tan (1998) [Rate control for communication networks: shadow prices, proportional fairness and stability, Journal of the Operational Research Society, 49(3), 237-252] involves fast user updates coupled with slow network updates in the form of additive-increase multiplicative-decrease of suggested user flows. The proposed algorithm may be viewed as one with fast user updates and fast network updates that keeps the iterates feasible at all times. Simulations suggest that the convergence rate of the ordinary differential equation (ODE) tracked by our proposed algorithms iterates is comparable to that of the ODE for the aforementioned benchmark algorithm.
This paper studies the distributed optimization problem where the objective functions might be nondifferentiable and subject to heterogeneous set constraints. Unlike existing subgradient methods, we focus on the case when the exact subgradients of the local objective functions can not be accessed by the agents. To solve this problem, we propose a projected primal-dual dynamics using only the objective functions approximate subgradients. We first prove that the formulated optimization problem can only be solved with an approximate error depending upon the accuracy of the available subgradients. Then, we show the exact solvability of this optimization problem if the accumulated approximation error is not too large. After that, we also give a novel componentwise normalized variant to improve the transient behavior of the convergent sequence. The effectiveness of our algorithms is verified by a numerical example.
In this paper, we present two completely uncoupled algorithms for utility maximization. In the first part, we present an algorithm that can be applied for general non-concave utilities. We show that this algorithm induces a perturbed (by $epsilon$) Markov chain, whose stochastically stable states are the set of actions that maximize the sum utility. In the second part, we present an approximate sub-gradient algorithm for concave utilities which is considerably faster and requires lesser memory. We study the performance of the sub-gradient algorithm for decreasing and fixed step sizes. We show that, for decreasing step sizes, the Cesaro averages of the utilities converges to a neighbourhood of the optimal sum utility. For constant step size, we show that the time average utility converges to a neighbourhood of the optimal sum utility. Our main contribution is the expansion of the achievable rate region, which has been not considered in the prior literature on completely uncoupled algorithms for utility maximization. This expansion aids in allocating a fair share of resources to the nodes which is important in applications like channel selection, user association and power control.
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