Subgradient methods for huge-scale optimization problems
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We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, which total cost depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions. We show that the updating technique can be efficiently coupled with the simplest subgradient methods, the unconstrained minimization method by B.Polyak, and the constrained minimization scheme by N.Shor. Similar results can be obtained for a new nonsmooth random variant of a coordinate descent scheme. We present also the promising results of preliminary computational experiments.
KeywordsNonsmooth convex optimization Complexity bounds Subgradient methods Huge-scale problems
Mathematics Subject Classification90C25 90C47 68Q25
The author would like to thank two the anonymous referees and associated editor for their very useful comments.
- 1.Khachiyan, L., Tarasov, S., Erlich, E.: The inscribed ellipsoid method. Sov. Math. Dokl. 37, 226–230 (1988)Google Scholar
- 5.Nesterov, Yu.: Efficiency of coordinate descent methods on huge-scale optimization problems. CORE iscussion paper 2010/2. Accepted by SIOPTGoogle Scholar
- 7.Gilpin, A., Peña, J., Sandholm, T.: First-order algorithm with \(O(\ln (1/\epsilon ))\) convergence for \(\epsilon \)-equilibrium in two-person zero-sum games. Math. Program. 133(2), 279–296 (2012)Google Scholar
- 8.Polyak, B.: Introduction to Optimization. Optimization Software, Inc., New York (1987)Google Scholar
- 9.Richtárik, P., Takac, M.: Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function. April 2011 (revised July 4, 2011). Math. Program. doi: 10.1007/s10107-012-0614-z