A variable smoothing algorithm for solving convex optimization problems

• Radu Ioan Bot ^[1] ; Christopher Hendrich ^[2]
  1. [1] University of Vienna

     University of Vienna

     Innere Stadt, Austria

     
  2. [2] Chemnitz University of Technology

     Chemnitz University of Technology

     Kreisfreie Stadt Chemnitz, Alemania

     
• Localización: Top, ISSN-e 1863-8279, ISSN 1134-5764, Vol. 23, Nº. 1, 2015, págs. 124-150
• Idioma: inglés
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• Resumen

  • In this article, we propose a method for solving unconstrained optimization problems with convex and Lipschitz continuous objective functions. By making use of the Moreau envelopes of the functions occurring in the objective, we smooth them to convex and differentiable functions with Lipschitz continuous gradients using both variable and constant smoothing parameters. The resulting problem is solved via an accelerated first-order method and this allows us to recover approximately the optimal solutions to the initial optimization problem with a rate of convergence of order O(lnkk) for variable smoothing and of order O(1k) for constant smoothing. Some numerical experiments employing the variable smoothing method in image processing and in supervised learning classification are also presented.