When is the Inverse of an Invertible Convex Function Itself Convex?

Robert Planqué

doi:10.68381/jca31041

Abstract

We provide a sufficient condition for an invertible (locally strongly) convex vector-valued function on

\mathbb{R}^N

to have a (locally strongly) convex inverse. We show under suitable conditions that if the gradient of each component of the inverse has negative entries, then this inverse is (locally strongly) convex if the original is.

Author information

Contact details are reproduced from the original publication and may be historical.

Robert Planqué

Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands

r.planque@vu.nl

Keywords

Invertible convex function
strongly convex function

Mathematics Subject Classification

26B25

References

F. J. Bruggeman, R. Planqué, D. Molenaar, B. Teusink: Searching for principles of microbial physiology, FEMS Microbiology Reviews 44 (2020) 821–844.https://doi.org/10.1093/femsre/fuaa034 Google Scholar
R. A. Horn, C. R. Johnson: Matrix Analysis, Cambridge University Press, Cambridge (1985).https://doi.org/10.1017/CBO9780511810817 Google Scholar
E. Noor, A. Flamholz, A. Bar-Even, D. Davidi, R. Milo, W. Liebermeister: The protein cost of metabolic fluxes: prediction from enzymatic rate laws and cost minimization, PLoS Comp. Biol. 12 (2016) e1005167.https://doi.org/10.1371/journal.pcbi.1005167 Google Scholar
R. Planqué, J. Hulshof, F. J. Bruggeman: How do bacteria control their metabolic state to maximise growth rate in varying environments?, forthcoming (2023).Google Scholar
R. Planqué, J. Hulshof, J. C. Hendriks, B. Teusink, F. J. Bruggeman: Maintaining maximal metabolic rate using gene expression control, PLoS Comp. Biol. 14 (2018) e1006412.https://doi.org/10.1371/journal.pcbi.1006412 Google Scholar
R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar
J. Smillie: Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Analysis 15 (1984) 530–534.https://doi.org/10.1137/0515040 Google Scholar
L. Thibault: Unilateral Variational Analysis in Banach Spaces. Part I: General Theory, World Scientific, Singapore (2023).https://doi.org/10.1142/12797 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2024. Rights now held by Banach Press.

[1] F. J. Bruggeman, R. Planqué, D. Molenaar, B. Teusink: Searching for principles of microbial physiology, FEMS Microbiology Reviews 44 (2020) 821–844.https://doi.org/10.1093/femsre/fuaa034 Google Scholar

[2] R. A. Horn, C. R. Johnson: Matrix Analysis, Cambridge University Press, Cambridge (1985).https://doi.org/10.1017/CBO9780511810817 Google Scholar

[3] E. Noor, A. Flamholz, A. Bar-Even, D. Davidi, R. Milo, W. Liebermeister: The protein cost of metabolic fluxes: prediction from enzymatic rate laws and cost minimization, PLoS Comp. Biol. 12 (2016) e1005167.https://doi.org/10.1371/journal.pcbi.1005167 Google Scholar

[4] R. Planqué, J. Hulshof, F. J. Bruggeman: How do bacteria control their metabolic state to maximise growth rate in varying environments?, forthcoming (2023).Google Scholar

[5] R. Planqué, J. Hulshof, J. C. Hendriks, B. Teusink, F. J. Bruggeman: Maintaining maximal metabolic rate using gene expression control, PLoS Comp. Biol. 14 (2018) e1006412.https://doi.org/10.1371/journal.pcbi.1006412 Google Scholar

[6] R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar

[7] J. Smillie: Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Analysis 15 (1984) 530–534.https://doi.org/10.1137/0515040 Google Scholar

[8] L. Thibault: Unilateral Variational Analysis in Banach Spaces. Part I: General Theory, World Scientific, Singapore (2023).https://doi.org/10.1142/12797 Google Scholar