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An infeasible-point subgradient method using adaptive approximate projections

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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.

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Acknowledgements

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) within the project “Sparse Exact and Approximate Recovery” under grants LO 1436/3-1 and PF 709/1-1. D. Lorenz further acknowledges support from the DFG project “Sparsity and Compressed Sensing in Inverse Problems” under grant LO 1436/2-1. Moreover, we thank the anonymous referees for their numerous helpful comments which greatly helped improving this paper.

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Authors and Affiliations

  1. Institute for Analysis and Algebra, TU Braunschweig, Pockelsstr. 14, 38106, Braunschweig, Germany

    Dirk A. Lorenz

  2. Research Group Optimization, TU Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Marc E. Pfetsch & Andreas M. Tillmann

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  1. Dirk A. Lorenz
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  2. Marc E. Pfetsch
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Correspondence to Andreas M. Tillmann.

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Lorenz, D.A., Pfetsch, M.E. & Tillmann, A.M. An infeasible-point subgradient method using adaptive approximate projections. Comput Optim Appl 57, 271–306 (2014). https://doi.org/10.1007/s10589-013-9602-3

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