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

Physics And Astronomy

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

Funding Information:
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 .

Publisher Copyright:
© 2015 Elsevier Inc.

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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N2 - 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.

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

Keywords

ASJC Scopus subject areas

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