Approximating convex functions via non-convex oracles under the relative noise model

Nir Halman

doi:10.1016/j.disopt.2014.12.001

Approximating convex functions via non-convex oracles under the relative noise model

Nir Halman

Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study succinct representations of a convex univariate function φ over a finite domain. We show how to construct a succinct representation, namely a piecewise-linear function φ¯ approximating φ when given a black box access to an L-approximation oracle φ∼of φ (the oracle value is always within a multiplicative factor L from the true value). The piecewise linear function φ¯ has few breakpoints (poly-logarithmic in the size of the domain and the function values) and approximates the true function φ up to a (1+∈)^L2 multiplicative factor point-wise, for any ∈>0. This function φ¯ is also convex so it can be used as a replacement for the original function and be plugged in algorithms in a black box fashion. Finally, we give positive and negative results for multivariate convex functions.

Original language	English
Pages (from-to)	1-16
Number of pages	16
Journal	Discrete Optimization
Volume	16
DOIs	https://doi.org/10.1016/j.disopt.2014.12.001
State	Published - May 2015
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2014 Elsevier B.V.

Funding

The author thanks Jim Orlin for fruitful discussions on an earlier version of this paper, and Chung-Lun Li for improving the presentation of the proof of Theorem 1.2 . The research was supported in part by the European Community’s Seventh Framework Programme FP7/2007–2013 [Grant agreement 247757] and the Recanati Fund of the School of Business Administration, the Hebrew University of Jerusalem.

Funders	Funder number
School of Business Administration
Seventh Framework Programme	247757
Hebrew University of Jerusalem

Keywords

Approximate binary search
Dynamic programming
Property preserving reconstruction

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10.1016/j.disopt.2014.12.001

Cite this

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Approximating convex functions via non-convex oracles under the relative noise model

Abstract

Bibliographical note

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Keywords

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