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Published June 2016 | Submitted
Journal Article Open

Bounding duality gap for separable problems with linear constraints

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.

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.

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