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Primal convergence from dual subgradient methods for convex optimization

Gustavsson, Emil, 1987 (author)
Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper, matematik,Department of Mathematical Sciences, Mathematics,University of Gothenburg,Chalmers tekniska högskola,Chalmers University of Technology
Patriksson, Michael, 1964 (author)
Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper, matematik,Department of Mathematical Sciences, Mathematics,Chalmers tekniska högskola,Chalmers University of Technology,University of Gothenburg
Strömberg, Ann-Brith, 1961 (author)
Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper, matematik,Department of Mathematical Sciences, Mathematics,University of Gothenburg,Chalmers tekniska högskola,Chalmers University of Technology
 (creator_code:org_t)
2014-04-16
2015
English.
In: Mathematical Programming. - : Springer Science and Business Media LLC. - 0025-5610 .- 1436-4646. ; 150:2, s. 365-390
  • Journal article (peer-reviewed)
Abstract Subject headings
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  • When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favorably utilized, since they often find near-optimal dual solutions quickly. However, an optimal primal solution is generally not obtained directly through such a subgradient approach unless the Lagrangian dual function is differentiable at an optimal solution. We construct a sequence of convex combinations of primal subproblem solutions, a so called ergodic sequence, which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen. We generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions. In contrast to previously proposed rules, they exploit more information from later subproblem solutions than from earlier ones. We evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed.

Subject headings

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)
NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Keyword

Convex programming
Ergodic convergence
Lagrangian duality
Nonlinear multicommodity flow problem
Primal recovery
Subgradient optimization
Ergodic convergence

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