Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We count as one operation the pair of operations formed by real multiplication and addition.
References
Khachiyan, L., Tarasov, S., Erlich, E.: The inscribed ellipsoid method. Sov. Math. Dokl. 37, 226–230 (1988)
Luo, Z.Q., Tseng, P.: On the convergence rate of dual ascent methods for linearly constrained convex minimization. Math. Oper. Res. 18(2), 846–867 (1993)
Nesterov, Yu.: Smooth minimization of non-smooth functions. Math. Program. A 103(1), 127–152 (2005)
Nesterov, Yu.: Primal-dual subgradient methods for convex problems. Math. Program. 120(1), 261–283 (2009)
Nesterov, Yu.: Efficiency of coordinate descent methods on huge-scale optimization problems. CORE iscussion paper 2010/2. Accepted by SIOPT
Nesterov, Yu., Nemirovskii, A.: Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia (1994)
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)
Polyak, B.: Introduction to Optimization. Optimization Software, Inc., New York (1987)
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
Shor, N.: Minimization Methods for Non-differentiable Functions. Springer Series in Computational Mathematics. Springer, Berlin (1985)
Acknowledgments
The author would like to thank two the anonymous referees and associated editor for their very useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research presented in this paper was partially supported by the Laboratory of Structural Methods of Data Analysis in Predictive Modeling, MIPT, through the RF government grant, ag.11.G34.31.0073.
Rights and permissions
About this article
Cite this article
Nesterov, Y. Subgradient methods for huge-scale optimization problems. Math. Program. 146, 275–297 (2014). https://doi.org/10.1007/s10107-013-0686-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-013-0686-4