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The Boosted Difference of Convex Functions Algorithm for nonsmooth functions

The Boosted Difference of Convex Functions Algorithm for nonsmooth functions
The Boosted Difference of Convex Functions Algorithm for nonsmooth functions
The Boosted Difference of Convex functions Algorithm (BDCA) was recently proposed for minimizing smooth difference of convex (DC) functions. BDCA accelerates the convergence of the classical Difference of Convex functions Algorithm (DCA) thanks to an additional line search step. The purpose of this paper is twofold. Firstly, to show that this scheme can be generalized and successfully applied to certain types of nonsmooth DC functions, namely, those that can be expressed as the difference of a smooth function and a possibly nonsmooth one. Secondly, to show that there is complete freedom in the choice of the trial step size for the line search, which is something that can further improve its performance. We prove that any limit point of the BDCA iterative sequence is a critical point of the problem under consideration, and that the corresponding objective value is monotonically decreasing and convergent. The global convergence and convergent rate of the iterations are obtained under the Kurdyka-Lojasiewicz property. Applications and numerical experiments for two problems in data science are presented, demonstrating that BDCA outperforms DCA. Specifically, for the Minimum Sum-of-Squares Clustering problem, BDCA was on average sixteen times faster than DCA, and for the Multidimensional Scaling problem, BDCA was three times faster than DCA.
Boosted difference of convex functions algorithm, Clustering problem, Difference of convex functions, Kurdyka–Lojasiewicz property, Multidimensional scaling problem
1052-6234
980-1006
Aragón Artacho, Francisco J.
27fd51f0-ca5f-4b38-86de-1b9c7eec8312
Vuong, Phan T.
52577e5d-ebe9-4a43-b5e7-68aa06cfdcaf
Aragón Artacho, Francisco J.
27fd51f0-ca5f-4b38-86de-1b9c7eec8312
Vuong, Phan T.
52577e5d-ebe9-4a43-b5e7-68aa06cfdcaf

Aragón Artacho, Francisco J. and Vuong, Phan T. (2020) The Boosted Difference of Convex Functions Algorithm for nonsmooth functions. SIAM Journal on Optimization, 30 (1), 980-1006. (doi:10.1137/18M123339X).

Record type: Article

Abstract

The Boosted Difference of Convex functions Algorithm (BDCA) was recently proposed for minimizing smooth difference of convex (DC) functions. BDCA accelerates the convergence of the classical Difference of Convex functions Algorithm (DCA) thanks to an additional line search step. The purpose of this paper is twofold. Firstly, to show that this scheme can be generalized and successfully applied to certain types of nonsmooth DC functions, namely, those that can be expressed as the difference of a smooth function and a possibly nonsmooth one. Secondly, to show that there is complete freedom in the choice of the trial step size for the line search, which is something that can further improve its performance. We prove that any limit point of the BDCA iterative sequence is a critical point of the problem under consideration, and that the corresponding objective value is monotonically decreasing and convergent. The global convergence and convergent rate of the iterations are obtained under the Kurdyka-Lojasiewicz property. Applications and numerical experiments for two problems in data science are presented, demonstrating that BDCA outperforms DCA. Specifically, for the Minimum Sum-of-Squares Clustering problem, BDCA was on average sixteen times faster than DCA, and for the Multidimensional Scaling problem, BDCA was three times faster than DCA.

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The Boosted DC Algorithm for nonsmooth functions - Accepted Manuscript
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More information

Accepted/In Press date: 24 January 2020
e-pub ahead of print date: 23 March 2020
Published date: 2020
Keywords: Boosted difference of convex functions algorithm, Clustering problem, Difference of convex functions, Kurdyka–Lojasiewicz property, Multidimensional scaling problem

Identifiers

Local EPrints ID: 437574
URI: http://eprints.soton.ac.uk/id/eprint/437574
ISSN: 1052-6234
PURE UUID: 230e1907-1c70-4cdf-8d8c-a65554eecd15
ORCID for Phan T. Vuong: ORCID iD orcid.org/0000-0002-1474-994X

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Date deposited: 06 Feb 2020 17:30
Last modified: 28 Apr 2022 02:28

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Contributors

Author: Francisco J. Aragón Artacho
Author: Phan T. Vuong ORCID iD

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