Approximate primal solutions and rate analysis for dual subgradient methods

Angelia Nedich, Asuman Ozdaglar

Research output: Contribution to journalArticle

175 Citations (Scopus)

Abstract

In this paper, we study methods for generati ng approximate primal solutions as a byproduct of subgradient methods applied to the Lagrangian dual of a primal convex (possibly nondifferentiable) constrained optimization problem. Our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods, such as the recent resource allocation problems in large-scale networks. For such problems, we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions. These algorithms use a constant stepsize in view of its simplicity and practical significance. We provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions. These estimates are given per iteration, thus providing a basis for analyzing the trade-offs between the desired level of error and the selection of the stepsize value. Our analysis relies on the Slater condition and the inherited boundedness properties of the dual problem under this condition. It also relies on the boundedness of subgradients, which is ensured by assuming the compactness of the constraint set.

Original languageEnglish (US)
Pages (from-to)1757-1780
Number of pages24
JournalSIAM Journal on Optimization
Volume19
Issue number4
DOIs
StatePublished - 2008
Externally publishedYes

Fingerprint

Subgradient Method
Dual Method
Dual Problem
Boundedness
Constrained optimization
Nondifferentiable Optimization
Resource allocation
Byproducts
Infeasibility
Subgradient
Constrained Optimization Problem
Efficient Implementation
Resource Allocation
Estimate
Averaging
Compactness
Simplicity
Optimal Solution
Trade-offs
Iteration

Keywords

  • Approximate primal solutions
  • Averaging
  • Convergence rate estimates
  • Subgradient methods

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

Cite this

Approximate primal solutions and rate analysis for dual subgradient methods. / Nedich, Angelia; Ozdaglar, Asuman.

In: SIAM Journal on Optimization, Vol. 19, No. 4, 2008, p. 1757-1780.

Research output: Contribution to journalArticle

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