Multiuser optimization: Distributed algorithms and error analysis

Jayash Koshal, Angelia Nedich, Uday V. Shanbhag

Research output: Contribution to journalArticle

54 Citations (Scopus)

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 user-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 step-length 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 step lengths 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.Analternative to primal-dual schemes can be found in dual schemes that 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 multiuser problem arising in a congested traffic network.

Original languageEnglish (US)
Pages (from-to)1046-1081
Number of pages36
JournalSIAM Journal on Optimization
Volume21
Issue number3
DOIs
StatePublished - 2011
Externally publishedYes

Fingerprint

Algorithm Analysis
Distributed Algorithms
Parallel algorithms
Error Analysis
Error analysis
Optimization Algorithm
Constrained optimization
Primal-dual
Dual space
Regularization Parameter
Utility Function
Choose
Gradient
Optimal Value Function
Traffic Network
Tikhonov Regularization
Linear Constraints
Constrained Optimization Problem
Approximation
Iterate

Keywords

  • Convex optimization
  • Distributed optimization
  • Gradient methods
  • Multiuser optimization
  • Projection methods
  • Variational inequalities

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software

Cite this

Multiuser optimization : Distributed algorithms and error analysis. / Koshal, Jayash; Nedich, Angelia; Shanbhag, Uday V.

In: SIAM Journal on Optimization, Vol. 21, No. 3, 2011, p. 1046-1081.

Research output: Contribution to journalArticle

Koshal, Jayash ; Nedich, Angelia ; Shanbhag, Uday V. / Multiuser optimization : Distributed algorithms and error analysis. In: SIAM Journal on Optimization. 2011 ; Vol. 21, No. 3. pp. 1046-1081.
@article{9e697d81ab554b55a65bab8d8d36ee35,
title = "Multiuser optimization: Distributed algorithms and error analysis",
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 user-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 step-length 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 step lengths 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.Analternative to primal-dual schemes can be found in dual schemes that 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 multiuser problem arising in a congested traffic network.",
keywords = "Convex optimization, Distributed optimization, Gradient methods, Multiuser optimization, Projection methods, Variational inequalities",
author = "Jayash Koshal and Angelia Nedich and Shanbhag, {Uday V.}",
year = "2011",
doi = "10.1137/090770102",
language = "English (US)",
volume = "21",
pages = "1046--1081",
journal = "SIAM Journal on Optimization",
issn = "1052-6234",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

}

TY - JOUR

T1 - Multiuser optimization

T2 - Distributed algorithms and error analysis

AU - Koshal, Jayash

AU - Nedich, Angelia

AU - Shanbhag, Uday V.

PY - 2011

Y1 - 2011

N2 - 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 user-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 step-length 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 step lengths 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.Analternative to primal-dual schemes can be found in dual schemes that 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 multiuser problem arising in a congested traffic network.

AB - 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 user-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 step-length 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 step lengths 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.Analternative to primal-dual schemes can be found in dual schemes that 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 multiuser problem arising in a congested traffic network.

KW - Convex optimization

KW - Distributed optimization

KW - Gradient methods

KW - Multiuser optimization

KW - Projection methods

KW - Variational inequalities

UR - http://www.scopus.com/inward/record.url?scp=80054752546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80054752546&partnerID=8YFLogxK

U2 - 10.1137/090770102

DO - 10.1137/090770102

M3 - Article

AN - SCOPUS:80054752546

VL - 21

SP - 1046

EP - 1081

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 3

ER -