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An infeasible-point subgradient method using adaptive approximate projections

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Listed:
  • Dirk Lorenz
  • Marc Pfetsch
  • Andreas Tillmann

Abstract

We propose a new subgradient method for the minimization of nonsmooth convex functions over a convex set. To speed up computations we use adaptive approximate projections only requiring to move within a certain distance of the exact projections (which decreases in the course of the algorithm). In particular, the iterates in our method can be infeasible throughout the whole procedure. Nevertheless, we provide conditions which ensure convergence to an optimal feasible point under suitable assumptions. One convergence result deals with step size sequences that are fixed a priori. Two other results handle dynamic Polyak-type step sizes depending on a lower or upper estimate of the optimal objective function value, respectively. Additionally, we briefly sketch two applications: Optimization with convex chance constraints, and finding the minimum ℓ 1 -norm solution to an underdetermined linear system, an important problem in Compressed Sensing. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Dirk Lorenz & Marc Pfetsch & Andreas Tillmann, 2014. "An infeasible-point subgradient method using adaptive approximate projections," Computational Optimization and Applications, Springer, vol. 57(2), pages 271-306, March.
    • Handle: RePEc:spr:coopap:v:57:y:2014:i:2:p:271-306
    DOI: 10.1007/s10589-013-9602-3
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    References listed on IDEAS

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    6. Willem Haneveld & Maarten Vlerk, 2006. "Integrated Chance Constraints: Reduced Forms and an Algorithm," Computational Management Science, Springer, vol. 3(4), pages 245-269, September.
    7. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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