Convexity of a small ball under quadratic map

Anatoly Dymarsky

doi:10.1016/j.laa.2015.09.005

Convexity of a small ball under quadratic map

Anatoly Dymarsky

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

4 Scopus citations

Abstract

We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak [1] in the case of a quadratic map. We also generalize the notion of the joint numerical range of m-tuple of matrices by adding vector-dependent inhomogeneous term and provide a sufficient condition for its convexity.

Original language	English
Article number	13356
Pages (from-to)	109-123
Number of pages	15
Journal	Linear Algebra and Its Applications
Volume	488
DOIs	https://doi.org/10.1016/j.laa.2015.09.005
State	Published - Jan 1 2016

Bibliographical note

Publisher Copyright:
© 2015 Elsevier Inc.

Funding

I would like to thank Tudor Dimofte, Konstantin Turitsyn and Jamin Sheriff for useful discussions and gratefully acknowledge support from the grant RFBR 15-01-04217 .

Funders	Funder number
Russian Foundation for Basic Research	15-01-04217

Keywords

Convexity
Joint numerical range
Quadratic transformation (map)
Trust region problem

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2015.09.005

Cite this

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AB - We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak [1] in the case of a quadratic map. We also generalize the notion of the joint numerical range of m-tuple of matrices by adding vector-dependent inhomogeneous term and provide a sufficient condition for its convexity.

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KW - Joint numerical range

KW - Quadratic transformation (map)

KW - Trust region problem

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Convexity of a small ball under quadratic map

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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