Abstract
The q-asymptotic function is a new tool that permits to study nonconvex optimization problems with unbounded data. It is particularly useful when dealing with quasiconvex functions. In this paper, we obtain formulas for the q-asymptotic function via c-conjugates, directional derivatives and subdifferentials. We obtain them under lower semicontinuity or local Lipschitz assumptions. The well-known formulas for the asymptotic function in the convex case are consequences of these ones. We obtain a new formula for the convex case.
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Notes
We assume that \(+\infty +(-\infty )=-\infty +(+\infty )=+\infty -(+\infty )=-\infty -(-\infty )=-\infty .\)
The definition of “\(\lim \sup \inf \)” is in the second line. We consider a simpler formula for \(f^{\uparrow }(x;\cdot )\) when f is lsc (see [12, Section 4]).
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Acknowledgements
The authors are grateful to the associate editor and two referees. Their valuable comments and suggestions contributed to a substantial improvement of the paper. This research was partially supported by Conicyt–Chile through Projects Fondecyt Postdoctorado 3160205 (Lara) and Fondecyt Regular 1150440 (López).
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Communicated by Constantin Zălinescu.
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Lara, F., López, R. Formulas for Asymptotic Functions via Conjugates, Directional Derivatives and Subdifferentials. J Optim Theory Appl 173, 793–811 (2017). https://doi.org/10.1007/s10957-017-1101-8
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DOI: https://doi.org/10.1007/s10957-017-1101-8
Keywords
- Asymptotic cones and functions
- q-Asymptotic functions
- c-Conjugates
- Directional derivatives
- Subdifferentials