Abstract
In this paper, theorems on existence and uniqueness of a vector minimizing a closed proper convex functionf are given, extending previous results on this topic. Necessary and sufficient conditions are presented for convex functions with additional properties. Iff attains its infimum atx withoutx being unique, linear functionsd'x with unique or bounded values are identified. Applicability of results is demonstrated in regression analysis of failure times, including the Cox model.
Zusammenfassung
In dieser Arbeit werden, für eine geschlossene eigentlich konvexe Funktionf, Existenz und Eindeutigkeit eines minimierenden Vektors untersucht und frühere Resultate zu diesem Thema verallgemeinert. Für konvexe Funktionen mit zusätzlichen Eigenschaften werden notwendige und hinreichende Bedingungen für Existenz bzw. Eindeutigkeit angegeben. Fallsf sein Minimum inx annimmt, ohne daßx eindeutig ist, werden lineare Funktionend'x mit eindeutigen oder beschränkten Werten identifiziert. Die Anwendbarkeit der Resultate wird in der Regressionsanalyse von Lebensdauern, etwa mit dem Cox Modell, demonstriert.
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Kaufmann, H. On existence and uniqueness of a vector minimizing a convex function. Zeitschrift für Operations Research 32, 357–373 (1988). https://doi.org/10.1007/BF01920035
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DOI: https://doi.org/10.1007/BF01920035