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Linear-step solvability of some folded concave and singly-parametric sparse optimization problems

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Abstract

This paper studies several versions of the sparse optimization problem in statistical estimation defined by a pairwise separation objective. The sparsity (i.e., \(\ell _0\)) function is approximated by a folded concave function; the pairwise separation gives rise to an objective of the Z-type. After presenting several realistic estimation problems to illustrate the Z-structure, we introduce a linear-step inner-outer loop algorithm for computing a directional stationary solution of the nonconvex nondifferentiable folded concave sparsity problem. When specialized to a quadratic loss function with a Z-matrix and a piecewise quadratic folded concave sparsity function, the overall complexity of the algorithm is a low-order polynomial in the number of variables of the problem; thus the algorithm is strongly polynomial in this quadratic case. We also consider the parametric version of the problem that has a weighted \(\ell _1\)-regularizer and a quadratic loss function with a (hidden) Z-matrix. We present a linear-step algorithm in two cases depending on whether the variables have prescribed signs or with unknown signs. In both cases, a parametric algorithm is presented and its strong polynomiality is established under suitable conditions on the weights. Such a parametric algorithm can be combined with an interval search scheme for choosing the parameter to optimize a secondary objective function in a bilevel setting. The analysis makes use of a least-element property of a Z-function, and, for the case of a quadratic loss function, the strongly polynomial solvability of a linear complementarity problem with a hidden Z-matrix. The origin of the latter class of matrices can be traced to an inspirational paper of Olvi Mangasarian to whom we dedicate our present work.

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Acknowledgements

The authors would like to thank Dr. Ying Cui at the University of Minnesota for her insightful comments that have helped to improve this manuscript, and for bringing to our attention the two references [16, 43]. They are also grateful to two referees who have provided constructive comments and offered additional references that have helped to improve the presentation and quality of the paper.

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  1. Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA, 90089, USA

    Andrés Gómez, Ziyu He & Jong-Shi Pang

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  1. Andrés Gómez
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The work of the first author was based on research support by the National Science Foundation under grant CIF-2006762. The work of the third author was based on research supported by the U.S. Air Force Office of Scientific Research under Grant FA9550-18-1-0382. A tribute to Olvi L. Mangasarian.

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Gómez, A., He, Z. & Pang, JS. Linear-step solvability of some folded concave and singly-parametric sparse optimization problems. Math. Program. 198, 1339–1380 (2023). https://doi.org/10.1007/s10107-021-01766-4

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