Subgradient methods in network resource allocation: Rate analysis

Nedić, Angelia; Ozdaglar, Asuman

doi:10.1109/ciss.2008.4558699

Cited by 27 publications

(12 citation statements)

References 22 publications

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“…The step-size of Assumption 3.1 is stronger than the more standard diminishing step-size scheme in [22] and this will correctly deal with the difficulty of the boundedness …”

Section: Distributed Subgradient Methodsmentioning

confidence: 99%

Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

Xiao¹

2016

ACM

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Abstract:In this paper, we present an improved subgradient algorithm for solving a general multi-agent convex optimization problem in a distributed way, where the agents are to jointly minimize a global objective function subject to a global inequality constraint, a global equality constraint and a global constraint set. The global objective function is a combination of local agent objective functions and the global constraint set is the intersection of each agent local constraint set. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. Our main focus is on constrained problems where the local constraint sets are identical. Thus, we propose a distributed primal-dual subgradient algorithm, which is based on the description of the primal-dual optimal solutions as the saddle points of the penalty functions. We show that, the algorithm can be implemented over networks with changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically converge to optimal solution with optimal value of the optimization problem under the Slater's condition.

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“…The step-size of Assumption 3.1 is stronger than the more standard diminishing step-size scheme in [22] and this will correctly deal with the difficulty of the boundedness …”

Section: Distributed Subgradient Methodsmentioning

confidence: 99%

Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

Xiao¹

2016

ACM

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“…A majority of past work on such problems (cf. [13,14]) requires that steplengths be consistent across users. Furthermore, in constrained regimes, there is a necessity to introduce both primal (user) steplengths and dual (link) steplengths.…”

mentioning

confidence: 99%

“…Both primal, primal-dual and dual schemes are discussed typically in a continuoustime setting (except for [13] where dual discrete-time schemes are investigated). Both dual and primal-dual discrete-time (approximate) schemes, combined with simple averaging, have been recently studied in [14][15][16] for a general convex constrained formulation. All of the aforementioned work establishes the convergence properties of therein proposed algorithms under the assumption that the users coordinate their steplengths, i.e., the steplength values are equal across all users.…”

mentioning

confidence: 99%

Multiuser Optimization: Distributed Algorithms and Error Analysis

2011

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Abstract. Traditionally, a multiuser problem is a constrained optimization problem characterized by a set of users, an objective given by a sum of user-specific utility functions, and a collection of linear constraints that couple the user decisions. The users do not share the information about their utilities, but do communicate values of their decision variables. The multiuser problem is to maximize the sum of the users-specific utility functions subject to the coupling constraints, while abiding by the informational requirements of each user. In this paper, we focus on generalizations of convex multiuser optimization problems where the objective and constraints are not separable by user and instead consider instances where user decisions are coupled, both in the objective and through nonlinear coupling constraints. To solve this problem, we consider the application of gradient-based distributed algorithms on an approximation of the multiuser problem. Such an approximation is obtained through a Tikhonov regularization and is equipped with estimates of the difference between the optimal function values of the original problem and its regularized counterpart. In the algorithmic development, we consider constant steplength primal-dual and dual schemes in which the iterate computations are distributed naturally across the users, i.e., each user updates its own decision only. Convergence in the primal-dual space is provided in limited coordination settings, which allows for differing steplengths across users as well as across the primal and dual space. We observe that a generalization of this result is also available when users choose their regularization parameters independently from a prescribed range. An alternative to primal-dual schemes can be found in dual schemes which are analyzed in regimes where approximate primal solutions are obtained through a fixed number of gradient steps. Per-iteration error bounds are provided in such regimes and extensions are provided to regimes where users independently choose their regularization parameters. Our results are supported by a case-study in which the proposed algorithms are applied to a multi-user problem arising in a congested traffic network.

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“…In , the authors proved that the average of { x ( l ) } converges to the primal optimal solution if σ ( l ) is set to a constant value

\overset{σ}{true}

throughout the subgradient iterations. Specifically,

\overset{bold-italicx}{falsemml-overline} MathClass-rel\to {bold-italicx}^{MathClass-bin*}

, where

\begin{array}{l} left & \overset{true¯}{boldnormalx} = \sum_{j = 1}^{l} {boldnormalx}^{(j)} / l \end{array}

…”

Section: Optimal Resource Allocation Algorithmmentioning

confidence: 99%

“…At the l th iteration, the primal constraints’ violations equal

B x = B {140%true \sum}^{​}_{l}^{j = 1} x^{(j)} ∕ l = \frac{π^{l + 1} - π^{1}}{l σ ̃}

p^{T} x - P = (p^{T} {140%true \sum}^{​}_{l}^{j = 1} x^{(j)} ∕ l - P \leq \frac{λ^{l + 1} - λ^{1}}{l σ ̃})

…”

Section: Optimal Resource Allocation Algorithmmentioning

confidence: 99%

Graph-based resource allocation algorithms for multiuser downlink MIMO-OFDMA networks

Zaki

Fapojuwo

2012

Wirel. Commun. Mob. Comput.

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Multiuser multiple‐input multiple‐output orthogonal frequency division multiple access (MIMO‐OFDMA) is considered as the practical method to attain the capacity promised by multiple antennas in the downlink direction. However, the joint calculation of precoding/beamforming and resource allocation required by the optimal algorithms is computationally prohibitive. This paper proposes computationally efficient resource allocation algorithms that can be invoked after the precoding and beamforming operations. To support stringent and diverse quality of service requirements, previous works have shown that the resource allocation algorithm must be able to guarantee a specific data rate to each user. The constraint matrix defined by the resource allocation problem with these data rate constraints provides a special structure that lends to efficient solution of the problem. On the basis of the standard graph theory and the Lagrangian relaxation, we develop an optimal resource allocation algorithm that exploits this structure to reduce the required execution time. Moreover, a lower‐complexity suboptimal algorithm is introduced. Extensive simulations are conducted to evaluate the computational and system‐level performance. It is shown that the proposed resource allocation algorithms attain the optimal solution at a much lower computational overhead compared with general‐purpose optimization algorithms used by previous MIMO‐OFDMA resource allocation approaches. Copyright © 2012 John Wiley & Sons, Ltd.

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Subgradient methods in network resource allocation: Rate analysis

Cited by 27 publications

References 22 publications

Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

Multiuser Optimization: Distributed Algorithms and Error Analysis

Graph-based resource allocation algorithms for multiuser downlink MIMO-OFDMA networks

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