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Bounding duality gap for separable problems with linear constraints

Udell, Madeleine and Boyd, Stephen (2016) Bounding duality gap for separable problems with linear constraints. Computational Optimization and Applications, 64 (2). pp. 355-378. ISSN 0926-6003. doi:10.1007/s10589-015-9819-4. https://resolver.caltech.edu/CaltechAUTHORS:20160620-094525529

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Abstract

We consider the problem of minimizing a sum of non-convex functions over a compact domain, subject to linear inequality and equality constraints. Approximate solutions can be found by solving a convexified version of the problem, in which each function in the objective is replaced by its convex envelope. We propose a randomized algorithm to solve the convexified problem which finds an ϵ -suboptimal solution to the original problem. With probability one, ϵ is bounded by a term proportional to the maximal number of active constraints in the problem. The bound does not depend on the number of variables in the problem or the number of terms in the objective. In contrast to previous related work, our proof is constructive, self-contained, and gives a bound that is tight.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s10589-015-9819-4DOIArticle
http://link.springer.com/article/10.1007/s10589-015-9819-4PublisherArticle
http://arxiv.org/abs/1410.4158arXivDiscussion Paper
ORCID:
AuthorORCID
Udell, Madeleine0000-0002-3985-915X
Additional Information:© 2016 Springer. Received 28 August 2014. Published online: 25 January 2016. The authors thank Haitham Hindi, Ernest Ryu and the anonymous reviewers for their very careful readings of and comments on early drafts of this paper, and Jon Borwein and Julian Revalski for their generous advice on the technical lemmas in the appendix.
Subject Keywords:Duality gap – Shapley–Folkman theorem – Separable problems – Convex relaxation – Randomized algorithms
Issue or Number:2
DOI:10.1007/s10589-015-9819-4
Record Number:CaltechAUTHORS:20160620-094525529
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160620-094525529
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:68519
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:20 Jun 2016 16:54
Last Modified:11 Nov 2021 04:01

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