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Primal convergence from dual subgradient methods for convex optimization

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Názov: Primal convergence from dual subgradient methods for convex optimization
Autori: Gustavsson, Emil, 1987, Patriksson, Michael, 1964, Strömberg, Ann-Brith, 1961
Zdroj: Mathematical Programming, Series B. 150(2):365-390
Predmety: Ergodic convergence, Lagrangian duality, Convex programming, Nonlinear multicommodity flow problem, Subgradient optimization, Primal recovery
Popis: 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.
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Prístupová URL adresa: https://research.chalmers.se/publication/205239
http://dx.doi.org/10.1007/s10107-014-0772-2
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  Label: Title
  Group: Ti
  Data: Primal convergence from dual subgradient methods for convex optimization
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Gustavsson%2C+Emil%22">Gustavsson, Emil</searchLink>, 1987<br /><searchLink fieldCode="AR" term="%22Patriksson%2C+Michael%22">Patriksson, Michael</searchLink>, 1964<br /><searchLink fieldCode="AR" term="%22Strömberg%2C+Ann-Brith%22">Strömberg, Ann-Brith</searchLink>, 1961
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  Label: Source
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  Data: <i>Mathematical Programming, Series B</i>. 150(2):365-390
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  Data: <searchLink fieldCode="DE" term="%22Ergodic+convergence%22">Ergodic convergence</searchLink><br /><searchLink fieldCode="DE" term="%22Lagrangian+duality%22">Lagrangian duality</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+programming%22">Convex programming</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+multicommodity+flow+problem%22">Nonlinear multicommodity flow problem</searchLink><br /><searchLink fieldCode="DE" term="%22Subgradient+optimization%22">Subgradient optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Primal+recovery%22">Primal recovery</searchLink>
– Name: Abstract
  Label: Description
  Group: Ab
  Data: 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.
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      – Type: doi
        Value: 10.1007/s10107-014-0772-2
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      – Text: English
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      Pagination:
        PageCount: 26
        StartPage: 365
    Subjects:
      – SubjectFull: Ergodic convergence
        Type: general
      – SubjectFull: Lagrangian duality
        Type: general
      – SubjectFull: Convex programming
        Type: general
      – SubjectFull: Nonlinear multicommodity flow problem
        Type: general
      – SubjectFull: Subgradient optimization
        Type: general
      – SubjectFull: Primal recovery
        Type: general
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      – TitleFull: Primal convergence from dual subgradient methods for convex optimization
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            NameFull: Gustavsson, Emil
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            NameFull: Patriksson, Michael
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            NameFull: Strömberg, Ann-Brith
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            – D: 01
              M: 01
              Type: published
              Y: 2015
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            – TitleFull: Mathematical Programming, Series B
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