Optimal subgradient algorithms for large-scale convex optimization in simple domains View Full Text


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Article Info

DATE

2017-03-14

AUTHORS

Masoud Ahookhosh, Arnold Neumaier

ABSTRACT

This paper describes two optimal subgradient algorithms for solving structured large-scale convex constrained optimization. More specifically, the first algorithm is optimal for smooth problems with Lipschitz continuous gradients and for Lipschitz continuous nonsmooth problems, and the second algorithm is optimal for Lipschitz continuous nonsmooth problems. In addition, we consider two classes of problems: (i) a convex objective with a simple closed convex domain, where the orthogonal projection onto this feasible domain is efficiently available; and (ii) a convex objective with a simple convex functional constraint. If we equip our algorithms with an appropriate prox-function, then the associated subproblem can be solved either in a closed form or by a simple iterative scheme, which is especially important for large-scale problems. We report numerical results for some applications to show the efficiency of the proposed schemes. More... »

PAGES

1071-1097

References to SciGraph publications

  • 2013-12-29. Bundle-level type methods uniformly optimal for smooth and nonsmooth convex optimization in MATHEMATICAL PROGRAMMING
  • 2014-05-28. Universal gradient methods for convex optimization problems in MATHEMATICAL PROGRAMMING
  • 2004. Introductory Lectures on Convex Optimization, A Basic Course in NONE
  • 2004-12-29. Smooth minimization of non-smooth functions in MATHEMATICAL PROGRAMMING
  • 2012-05-05. Fine tuning Nesterov’s steepest descent algorithm for differentiable convex programming in MATHEMATICAL PROGRAMMING
  • 2009-02-06. Primal-dual first-order methods with iteration-complexity for cone programming in MATHEMATICAL PROGRAMMING
  • 2011-05-09. Proximal Splitting Methods in Signal Processing in FIXED-POINT ALGORITHMS FOR INVERSE PROBLEMS IN SCIENCE AND ENGINEERING
  • 2015-05-16. OSGA: a fast subgradient algorithm with optimal complexity in MATHEMATICAL PROGRAMMING
  • 2011-07-30. Templates for convex cone problems with applications to sparse signal recovery in MATHEMATICAL PROGRAMMING COMPUTATION
  • 2010-12-21. A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging in JOURNAL OF MATHEMATICAL IMAGING AND VISION
  • 2014. Introduction to Nonsmooth Optimization, Theory, Practice and Software in NONE
  • 2007-10. Regularization Tools version 4.0 for Matlab 7.3 in NUMERICAL ALGORITHMS
  • 2013-06-14. First-order methods of smooth convex optimization with inexact oracle in MATHEMATICAL PROGRAMMING
  • 2013-06-10. An inexact line search approach using modified nonmonotone strategy for unconstrained optimization in NUMERICAL ALGORITHMS
  • 2012-12-21. Gradient methods for minimizing composite functions in MATHEMATICAL PROGRAMMING
  • 2012-12-13. A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems in COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
  • 2017-04-12. An optimal subgradient algorithm for large-scale bound-constrained convex optimization in MATHEMATICAL METHODS OF OPERATIONS RESEARCH
  • 1995-07. New variants of bundle methods in MATHEMATICAL PROGRAMMING
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    http://scigraph.springernature.com/pub.10.1007/s11075-017-0297-x

    DOI

    http://dx.doi.org/10.1007/s11075-017-0297-x

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