Abstract
If K(t) are sets of admissible solutions in parametric programs then it is natural to ask about the Lipschitz-like property and the lower semi-continuity of the multifunction. Answers to this question are related to the problem of the continuity or Lipschitz continuity of the value function, namely having the lower semi-continuity of K(·) we get the upper semi-continuity of the function easily and the Lipschitz-like property of K(·) leads to the Lipschitz-continuity of it. Herein sufficient conditions to get these properties of the polyhedral multifunction of admissible solutions are given in terms of the lower limit of the Hoffman constant. It is shown that the multifunction is Lipschitz-like at these parameters at which the lower limit of the Hoffman constant are positive.
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This paper was written while the second author was visiting Professor at the University of Burgundy. The author would like to thank the authority of the University for the hospitality.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Jourani, A., Zagrodny, D. The positiveness of lower limits of the Hoffman constant in parametric polyhedral programs. J Glob Optim 53, 641–661 (2012). https://doi.org/10.1007/s10898-011-9729-7
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DOI: https://doi.org/10.1007/s10898-011-9729-7
Keywords
- Parametric programming
- Hoffman constant
- Error bound
- Mosco convergence
- Attouch’s theorem
- Convex functions
- Subdifferentials
- Polyhedrals